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Pseudospectral methods on a semi-infinite interval with application to the Hydrogen atom: a comparison of the mapped Fourier-sine method with Laguerre series and rational Chebyshev expansions
Abstract
The Fourier-sine-with-mapping pseudospectral algorithm of Fattal et al. [Phys. Rev. E 53 (1996) 1217] has been applied in several quantum physics problems. Here, we compare it with pseudospectral methods using Laguerre functions and rational Chebyshev functions. We show that Laguerre and Chebyshev expansions are better suited for solving problems in the interval r∈[0,∞] (for example, the Coulomb–Schrödinger equation), than the Fourier-sine-mapping scheme. All three methods give similar accuracy for the hydrogen atom when the scaling parameter L is optimum, but the Laguerre and Chebyshev methods are less sensitive to variations in L. We introduce a new variant of rational Chebyshev functions which has a more uniform spacing of grid points for large r, and gives somewhat better results than the rational Chebyshev functions of Boyd [J. Comp. Phys. 70 (1987) 63].
... Pseudospectral methods in general are discussed in [60,32,13]. The author has previously explained in [7,23] why rational Chebyshev basis functions are good for the semi-infinite interval because of their connection with ordinary Fourier series and simplicity of programming. ...
... The pseudospectral algorithm is described in [7,23]. The Maple code is given in the appendix of [22]. ...
... A number of remedies immediately suggest themselves such as 1. Applying the asymptotic approximation for large Z and matching to a spectral expansion on a large but finite domain; 2. Replacing the rational Chebyshev basis by Laguerre functions; 3. Replacing the rational Chebyshev basis by the TM basis of [23]; 4. Replacing the sinh map by a more complicated change of coordinates with slower growth for very large Z ; 5. Applying perturbation theory such as the delta expansion [4,20], and sundry uncatalogued strategies. Because there are many options, resolution of the sinh map difficulties really requires a separate paper. ...
We revisit the spectral solution of the Thomas–Fermi problem for neutral atoms, urr−(1/r)u3/2=0 on r∈[0,∞] with u(0)=1 and u(∞)=0 to illustrate some themes in solving differential equations when there are complications, and also to make improvements in our earlier treatment. By “complications” we mean features of the problem that either destroy the exponential accuracy of a standard Chebyshev series, or render the classic Chebyshev approach inapplicable. The Thomas–Fermi problem has four complications: (i) a semi-infinite domain r∈[0,∞] (ii) a square root singularity in u(r) at the origin (iii) a fractional power nonlinearity and (iv) asymptotic decay as r→∞ that includes negative powers of r with fractional exponents. Our earlier treatment determined the slope at the origin to twenty-five decimal places, but no fewer than 600 basis functions were required to approximate a univariate solution that is everywhere monotonic, and all of the earlier tricks failed to recover an exponential rate of convergence in the truncation of the spectral series N, but only a high order convergence in negative powers of N. Here, using the coordinate z≡r to neutralize the square root singularity as before, we show that accuracy and rate of convergence are significantly improved by solving for the original unknown u(r) instead of the modified unknown v(r)=u(r) used previously. Without a further change of coordinate, a rational Chebyshev basis TLn(z;L) yields twelve decimal place accuracy for the slope at the origin, ur(0), with 70 basis functions and twenty-four decimal places with a truncation N=100. True exponential accuracy can be restored by using an appropriate change of coordinate, z=G(Z), where G is some species of exponential. However, the various Chebyshev and Fourier series for the Thomas–Fermi function have “plural asymptotics” that is, an∼aintermediate(n) for 1«n«nI but an∼afar(n) for n≫nI for some positive constant nI. The “far-asymptotics” as n→∞ are often of no practical significance. For this problem, the TL-with-sinh method, theoretically the best for huge n, is bedeviled with numerical ill-conditioning and its asymptotic superiority is realized only when the goal is at least forty decimal places of accuracy. This is absurd for engineering, but useful perhaps for benchmarking. To sixty decimal places, ur(0)=−1.588071022611375312718684509423950109452746621674825616765677. To nineteen digits after the decimal point, the constant in the Coulson–March asymptotic series is improved to F=13.2709738480269351535.
... The critical loads for divergence of a clamped-free elastic bar of finite length l and constant flexural rigidity α exposed to an end load Q are determined by the fourth order non-self-adjoint boundary eigenvalue problem ⎧ ⎨ ⎩ y (4) (x) = λy (2) (x) , 0 < x < 1, y (0) = y (0) = 0, y (2) (1) = 0, y (3) ...
... Consequently, we will also consider the boundary λ-independent eigenvalue problem attached to (1), namely ⎧ ⎨ ⎩ y (4) (x) = λy (2) (x) , 0 < x < 1, y (0) = y (0) = 0, y (2) (1) = 0, γy (3) (1) + (1 − γ ) y (3) ...
... Consequently, we will also consider the boundary λ-independent eigenvalue problem attached to (1), namely ⎧ ⎨ ⎩ y (4) (x) = λy (2) (x) , 0 < x < 1, y (0) = y (0) = 0, y (2) (1) = 0, γy (3) (1) + (1 − γ ) y (3) ...
In this paper, we consider the numerical treatment of singular eigenvalue problems supplied with eigenparameter dependent boundary conditions using spectral methods. On the one hand, such boundary conditions hinder the construction of test and trial space functions which could incorporate them and thus providing well-conditioned Galerkin discretization matrices. On the other hand, they can generate surprising behavior of the eigenvectors hardly detected by analytic methods. These singular problems are often indirectly approximated by regular ones. We argue that spectral collocation as well as tau method offer remedies for the first two issues and provide direct and efficient treatment to such problems. On a finite domain, we consider the so-called Petterson-König’s rod eigenvalue problem and on the half line, we take into account the Charney’s baroclinic stability problem and the Fourier eigenvalue problem. One boundary condition in these problems depends on the eigenparameter and additionally, this also could depend on some physical parameters. The Chebyshev collocation based on both, square and rectangular differentiation and a Chebyshev tau method are used to discretize the first problem. All these schemes cast the problems into singular algebraic generalized eigenvalue ones which are solved by the QZ and/or Arnoldi algorithms as well as by some target oriented Jacobi-Davidson methods. Thus, the spurious eigenvalues are completely eliminated. The accuracy of square Chebyshev collocation is roughly estimated and its order of approximation with respect to the eigenvalue of interest is determined. For the problems defined on the half line, we make use of the Laguerre-Gauss-Radau collocation. The method proved to be reliable, accurate, and stable with respect to the order of approximation and the scaling parameter.
... In the next section, we describe how to solve eigenproblems on a semi-infinite domain using an orthodox pseudospectral method. Boyd, Rangan and Buchsbaum [3] proposed an alternative basis; in Section 4, we explain why the ''TM'' basis is sometimes better in principle, and also show that it is indeed slightly superior to the ''TL'' basis for this particular application. ...
... Boyd, Rangan and Buchsbaum [3] introduced a new family of basis functions for the interval y ∈ [0, ∞]. Like the rational Chebyshev functions TL n (x; L) introduced by Boyd [16], the TM n (x; L M ) functions are the images of Chebyshev polynomials under a one-parameter family of mappings. ...
... This is a desirable property when the solution decays exponentially with y: no point in putting interpolation points where the function being interpolated has no amplitude. For the quantum eigenproblems of [3], the TM basis gave ''somewhat better results''. How do the basis sets compare for a hydrodynamic instability problem? ...
The Charney problem, a second order ordinary differential equation eigenproblem on with complex eigenvalues, is of great historical importance in meteorology and oceanography. Here, it is used as a testbed for several extensions of spectral methods. The first is to parameterize a plane curve which is singular at an endpoint, as very common in applications. The second stretch is to extend the Chebyshev tau method to compute eigenfunctions of the form where and are entire functions and where the approximation interval is a line segment in the complex plane. Third, we offer a special procedure for finding the roots of a function which is not a polynomial, but rather the combination of a polynomial plus a logarithm multiplied by a second polynomial. Lastly, to resolve the very thin boundary layer of the regularized Charney problem, we combine a rational Chebyshev ( ) pseudospectral method with a change of coordinate which is quadratic at the ground. Remarkably, best results are obtained by applying four boundary conditions even though the Charney problem is a differential equation of only second order.
... Thus, a good heuristic (empirical), but otherwise general strategy, to optimize the scaling parameter is to simply solve the problem and analyze the spectral coefficients of the expansion in the phase space, plotted as absolute values on a log-linear plot, for several different values of the scaling parameter. This strategy has been introduced by Boyd and his coworkers in [4] (see also [16]) and will be exploited below. It enables a clear separation of truncation and roundoff errors as well as a rough estimation of the global accuracy. ...
... In order to try to explain this difference in accuracy we take over the idea from [4] and represent the absolute values of the coefficients of solutions expansions in phase space versus the order of approximation N. As with any collocation method we effectively work in the physical space and consequently we need some polynomial transformations in order to transfer the solution in the frequency (phase) space. Actually, we use the fast Chebyshev transform from [31] and the discrete Laguerre transform from [29]. ...
... It is clear that for b = 15 in this region the coefficients of the Laguerre expansion are essentially random numbers. Consequently, for expansions on an unbounded interval we confirm the assertion from [4] that the spectral Laguerre coefficients oscillate in degree as well as decay but one can usually bound them from above by a straight line, the so called the envelope. The error estimate is then the magnitude of the envelope at the truncation limit. ...
Three spectral collocation methods, namely Laguerre collocation (LC), Laguerre Gauss Radau collocation (LGRC) and mapped Chebyshev collocation (ChC) are used in order to solve some challenging systems of boundary layer problems of third and second orders. The last two methods enable a Fourier type analysis, mainly (fast) polynomial transformations, which can be used in order to improve the process of optimization of the scaling parameters. Generally, the second method mentioned above produces the best results. Unfortunately they remain sub geometric with respect to the accuracy. However, all methods avoid domain truncation and rather arbitrary shooting techniques. Some challenging problems from fluid mechanics, including non-newtonian fluids are accurately solved.
... An alternative approach to optimise the distribution of the grid points is using a mapping function to transform the physical coordinate(s) to the computational coordinate(s) [29,18,[30][31][32][33], without finding the true eigenstates in advance like in the PO-DVR method. This method is more flexible since the mapping function can be chosen freely. ...
... In order to improve the efficiency of the FE-DVR method, one may have two choices: One is to adopt a flexible mapping function to improve the distribution of the grid points and the other one is develop new DVR other than the LDVR by using suitable quadrature rule and corresponding polynomials [31,62]. In our previous work, a mapped FE-DVR method was developed for improving the grid distribution of the original FE-DVR method, which improves the grid distribution efficiency significantly, especially for the problems involving long-range interactions [63]. ...
... Many numerical methods have been proposed to deal with it. The LDVR method [60], the FE-DVR method [45], the Coulomb function DVR method [72], the rational Chebyshev method [31], the Gauss-Legendre DVR [62], the mapped Fourier [19] and Sinc-DVR methods [70] are among the best ones. We will shown below that the OLDVR and OFE-DVR methods can significantly improve the efficiency of the grid points, as comparing with the original LDVR and FE-DVR methods for this Coulomb singularity in spherical coordinates. ...
... In order to try to explain this difference in accuracy we borrow a feasible idea from [34]. It means to represent in a log-linear plot (semilogy in MATLAB) the absolute To this aim we use the fast Chebyshev transform from [192] and the discrete Laguerre transform from [187] (see also [163] Sect. ...
... Consequently, for expansions on an unbounded interval, we confirm the assertion from [34], namely the spectral Laguerre coefficients oscillate in degree as well as decay but one can usually bound them from above by a straight line, the so called the envelope. This line parallelizes the straight line through origin ...
\begin{quotation}
\textit{If the numerical analysis is an art and not merely
a science, it is hoped that this art show will illustrate
the little tricks that are important in applications.}\\
\textbf{J. P. Boyd}, 1987
\end{quotation}
The foundations of spectral methods are not recent. For many years, before the appearance of
contemporary computers, the theoretical studies in mathematical physics, and especially in fluid mechanics,
made an extensive use of series expansions using the so-called "special functions".
The main aim of this work is to show that spectral collocation, based on
Laguerre, Hermite and Sinc functions can efficiently and reliably solve initial and/or boundary
value problems, as well as eigenvalue problems, formulated on unbounded domains. The expected convergence rates are $O\left(e^{-cN}\right)$ or
at least $O\left(e^{-c\sqrt{N}}\right)$ where $N$ is the number of degrees of freedom in the expansion of solution,
i.e., the approximation order, and $c$ is a constant independent of $N$. These preoccupations on spectral methods are
originated in our previous text \cite{CIG07}.
A major advantage in using these basis functions consists in the fact that for such problems boundary conditions are
usually “natural” rather than “essential” in the sense that the singularities of the differential
problem will force the numerical solution to
have the correct behavior at infinity even if no constraints are imposed on these functions.
In this way we definitely refuse the use of empirical \textit{domain truncation} along with \textit{various
shooting techniques}.
We mainly focus on the computational and practical aspects of this type of collocation. We do not
prove theorems concerning analytical aspects of this method but we carry out a lot of numerical
experiments and then analyze and compare their outcomes.\footnote{\label{WMK} And it is
not uncommon for proofs to be wrong. Philip Davis has done a survey in mathematical proofs
and found that something in the order of third of them have serious deficiencies-a tenth of them,
uncorrectable-but, thank goodness, nobody cares about most of those theorems.
Therefore you also have to have \textit{\textbf{testing}}. Testing requires a level of
deviousness and ingenuity and understanding that transcends what is needed to write the
programm in the first place. William M. Kahan, 2005 (see: \texttt{history.siam.org/})}
Accordingly we utilize these numerical investigations in order to find \textit{the goods and the
bads} of spectral collocation applied to various problems formulated on unbounded domains.
Thus, in the first Chapter we carry out a review of the main weighted interpolation
techniques available for unbounded intervals. We consider Hermite and Laguerre
interpolations based on homonymous functions as well as the interpolation
based on Sinc functions. Some mapping techniques, which bring into play the
best approximation polynomials, i.e., the Chebyshev polynomials, are also
reviewed along with some elementary preconditioning. Technical aspects concerning polynomial transforms and boundary condition implementation
are also shortly examined.
Most important, we make use of \textit{a strategy to evaluate the convergence rate} (the accuracy) of our collocation solutions.
Namely, from the physical space, where every collocation works, we transfer the solution into the coefficient (phase) space.
In this aim we use FFT and polynomial transforms. Then, in a log linear plot, we analyse (measure) the rate at which the expansion
coefficients of solution decrease to zero. In this way a rough estimation of the accuracy becomes possible.
This concept is graphically illustrated.
In the second Chapter we use LC and LGRC in order to solve boundary value problems on the half line and
HC and SiC to solve problems on the real line. We successively consider a linear test problem,
a class of nonlinear second order t. p. b. v. p.,
the modified Troesch's and Thomas-Fermi's problems, a semiconductor problem and the Kidder's problem.
The Falkner-Skan equation coupled with the heat transfer equation and the Blasius viscous flow of a kind of a
non-Newtonian fluid are briefly reviewed. In order to assess the \textit{accuracy} of our outcomes we
use the strategy introduced in the first Chapter. For a linear test problem
on the real line we determine the order of convergence of SiC.
In the third Chapter we deal with some singular eigenvalue problems as well as with
eigenvalue problems supplied with boundary conditions depending on the
eigenparameter. As a typical example of eigenvalue problem with eigenparameter dependent boundary conditions we have
chosen the so called Fourier problem. This is a second order
differential problem well studied analytically. This has enabled us some
important validation of our numerical results. The second problem is the
so called Charney's baroclinic stability problem which involves an analytic
singularity.
The Schr\"{o}dinger eigenvalue problems on the half as well as on the real line occupy
a large space in this Chapter. We attach various potentials, bounded or unbounded, to these problems.
A particular attention is paid to the challenging issue of \textit{continuous spectra} and
numerical (discrete) eigenvalues. It is known that some spectral problems for differential operators which are
naturally posed on the whole real line,
often lead to discrete eigenvalues plus a continuous spectrum. Any numerical approximation
typically involves three processes: (a) reduction to a finite interval; (b) discretization;
(c) application of a numerical eigenvalue solver.
Reduction to a finite interval and discretization typically eliminates the continuous
spectrum. Even if we do not truncate, a priori and arbitrary, the domain on which the problem is
formulated, such an inherent reduction can not be avoided.
Two techniques are used in order to eliminate the "bad" eigenvalues as well as to estimate the accuracy
of computations. The first one is the \textit{drift}, with respect to the order of approximation
and the scaling factor, of some eigenvalues of interest. The second one is based on the check of the \textit{eigenvectors orthogonality}.
We have slightly extended the notion of drift introduced by John Boyd
and confirmed its validity in appreciating the correctness of computed eigenvalues.
The algebraic generalized eigenvalue problems involved have been solved by various methods including
some subspace methods of Jacobi-Davidson type.
Actually in this Chapter we carry on the main topic of our recent monograph \textit{Spectral Methods
for Non Standard Eigenvalue Problems. Fluid and Structural Mechanics and Beyond}, Springer, 2014.
In the introductory part of the fourth Chapter we consider some linear multidimensional boundary value problems
which can be reduced to one dimensional t. p. b. v. p. (various self-similar solutions to quasilinear parabolic equations).
The main part of this Chapter is devoted to
\textit{weakly nonlinear p. d. e. of evolution}. The Benjamin Bona Mahony (BBM),
modified BBM and Benjamin Bona Mahony-Burgers (BBM-B),
the K\"{o}rteweg-de Vries (KdV), and the nonlinear Schr\"{o}dinger (NLS) initial value problems on the real axis are important
examples of such problems. All these equations involve linear dispersion.
For BBM and BBM-B cases we have some preliminary results in our recent contributions.
In order to solve such
problems we use the \textit{method of lines}, i.e., we couple various high order
collocation methods for the discretization of spatial variables with some finite difference schemes in order to march in time.
Then, we prove the linear stability of the method. In order to determine the regions of
linear stability we use the pseudospectra of the discrete linearized operators (matrices).
Particular radially symmetric solutions (waveguide solutions) as well as general radially symmetric solutions
to NLS have been computed. In the latter case we take into account Gaussian and Lorentzian initial data. Blow-up
self similar solutions to the cubic NLS and their convergence to Townes solitons is also examined.
Additionally, we pay attention to accurate approximation of various bound states of multiple
"solitons" solutions to cubic NLS.
Whenever there exist invariants of the above problems, a special attention is paid to their conservation.
The importance of such invariants is two folded. First, we can use them in order to optimize the
scaling parameter. Second, they provide useful hints in order to eliminate spurious solutions of
some nonlinear boundary value problems solved by collocation.
Whenever, a boundary value problem exhibits some \textit{first integrals} we can use the \textit{modal persistence} or alternatively
the $\rho$-\textit{criterion} in order to assure the numerical stability and to eliminate the spurious solutions.
In the fifth Chapter we provide some typical MATLAB scripts. They could improve the understanding
of the text and could open the way for various numerical experiments.
We end up with a Concluding Remarks Chapter. The existence of a
scaling parameter in all interpolation techniques considered, enables one to avoid
the empirical domain truncation and to adjust the numerical outcomes. By trial and error we can improve the
accuracy, stability and the conservation of various invariants. Unfortunately a rigorous and
automatic way to perform the tuning of scaling parameters is still missing.
A Section containing the acronyms used throughout the work has been provided in order to facilitate the reading.
Throughout this work the numerical results have been carried out using the MATLAB
environment on a HP xw8400 workstation with clock speed of 3.2 Ghz.
The author acknowledges the friendly and lucrative climate from "Tiberiu Popoviciu"
Institute of Numerical Analysis, Cluj-Napoca, Romania, which has decisively contributed to
the accomplishing of this volume.
Last but not least, I will fairly appreciate any remark, suggestion and correction concerning this text. These will be a relevant indicator
of this open project, and I promise to address them all and incorporate them in the future editions.
\vspace{\baselineskip}
\begin{flushright}\noindent
Cluj-Napoca, Romania \hfill {\it C\u{a}lin-Ioan Gheorghiu}\\
January 2018 \hfill \\
\end{flushright}
... Parity acceleration cannot be applied to basis sets that lack parity such as Laguerre functions [16,9,12,6,22,23] and the rational Chebyshev functions on the half-line [3,17,9,6]. ...
... Parity acceleration cannot be applied to basis sets that lack parity such as Laguerre functions [16,9,12,6,22,23] and the rational Chebyshev functions on the half-line [3,17,9,6]. ...
... The DVR of the generalised Laguerre polynomials only is able to describe a subset of the eigenstate of a Coulomb potential at a time and they are not suitable for describing the ionisation continua. [18,19] The Lobatto-DVR can accurately represent the Coulomb singularity in spherical coordinates, with unnecessarily dense grid points at both ends of the grid, [17,20] where the Lobatto shape functions on Gauss-Lobatto quadrature are taken as the basis. [17,21,22] The Coulomb DVR works well with a single Coulomb singularity in spherical coordinates, where its corresponding basis is the Coulomb wave functions. ...
... It is very possible that the FD methods, in a combination with certain variable mapped functions, would become much more efficient, similar to that for the Sinc DVR and Fourier method. [19,[68][69][70][71] In that case the spectral FD methods may be very useful also for treating with the Coulomb potential. [33,72,73] This worth more investigation in the future work. ...
We proposed a distributed approximating functional method for efficiently describing the electronic dynamics in atoms and molecules in the presence of the Coulomb singularities, using the kernel of a grid representation derived by using the solutions of the Coulomb differential equation based upon the Schwartz's interpolation formula, and a grid representation using the Lobatto/Radau shape functions. The elements of the resulted Hamiltonian matrix are confined in a narrow diagonal band, which is similar to that using the (higher order) finite difference methods. However, the spectral convergence properties of the original grid representations are retained in the proposed distributed approximating functional method for solving the Schr\"odinger equation involving the Coulomb singularity. Thus the method is effective for solving the electronic Schr\"odinger equation using iterative methods where the action of the Hamiltonian matrix on the wave function need to evaluate many times. The method is investigated by examining its convergence behaviours for calculating the electronic states of the H atom, H$_2^+$ molecule, the H atom in a parallel magnetic and electric fields, as the radial basis functions.
... Spectral and collocation methods are discussed in a famous monograph by D. Gottlieb and S. Orszag [94]. In atomic and molecular physics, collocation methods have been used to study a great variety of systems, using various sets of interpolation functions [191,36,161,35,67,37]. Multidimensional grids have been implemented to study systems with more than one degree of freedom [127]. ...
... In Ref. [276] we discussed the appearance of spurious solutions in calculations of weakly bound vibrational levels of diatomic molecules. Spurious solutions arising in pseudo-spectral methods were also reported by J. P. Boyd et al. [37]. ...
Thèse dans le cadre d'une convention de co-tutelle entre l'Université Paris-Sud XI (Orsay) et l'Université de Hanovre (Allemagne)
... Numerical solution of differential equations can also be obtained with the pseudospectral (PS) method which is based on a combination of the spectral and real-space representations. It has been found to be very accurate in solving onedimensional Schrödinger equation [35][36][37][38][39][40], in particular the Kohn-Sham (KS) equation for atoms [41,42] described with the density functional theory (DFT). The KS equation includes only the local (multiplicative) effective potential which makes the application of the PS method straightforward once suitable scaling is applied. ...
The Hartree–Fock (HF) equation for atoms with closed (sub)shells is transformed with the pseudospectral (PS) method into a discrete eigenvalue equation for scaled orbitals on a finite radial grid. The Fock exchange operator and the Hartree potential are obtained from the respective Poisson equations also discretized using the PS representation. The numerical solution of the discrete HF equation for closed-(sub)shell atoms from He to No is robust, fast and gives extremely accurate results, with the accuracy superior to that of the previous HF calculations. A very moderate number of 33 to 71 radial grid points is sufficient to obtain total energies with 14 significant digits and occupied orbital energies with 12 to 14 digits in numerical calculations using the double precision (64-bit) of the floating-point format.The electron density at the nucleus is then determined with 13 significant digits and the Kato condition for the density and s orbitals is satisfied with the accuracy of 11 to 13 digits. The node structure of the exact HF orbitals is obtained and their asymptotic dependence, including the common exponential decay, is reproduced very accurately. The accuracy of the investigated quantities is further improved by performing the PS calculations in the quadruple precision (128-bit) floating-point arithmetic which provides the total energies with 25 significant digits while using only 80 to 130 grid points.
... These two functions are mutually orthogonal on the semiinfinite domain. Therefore, subsequent research was shifted towards solving the nonlinear ordinary differential equations on semi-infinite intervals by applying these rational functions in spectral methods such as collocation and Tau approach based on operational matrix [21][22][23][24][25][26][27][28][29][30]. ...
Two new functions on the semi-infinite interval, namely Rational Gegenbauer (R) and Exponential Gegenbauer (E) functions are proposed to solve the heat transfer problem. The considered problem is flow of MHD micropolar over a moving plate with suction and injection boundary conditions. For applying Tau method efficiently, two matrices of derivative and product for both of rational and exponential Gegenbauer whose enable us to solve a system of nonlinear algebraic equations on the semi-infinite interval were introduced, and an error bound of these functions approximation was estimated which led to have an exponential convergence rate in this method. Moreover, the influence of the important physical parameters on heat and mass transfer phenomena are studied with details. Comparing the results of Rational Gegenbauer Tau and Exponential Gegenbauer Tau methods with available analytical and numerical solutions shows that the present methods are efficient and have fast convergence rate and high accuracy. This method can solve a set of coupled nonlinear and high-order differential equations on a semi-infinite domain by converting to a set of linear equations.
... 9,[13][14][15][16][17] In various combinations, these ideas have all been applied to the electronic structure problem in the past. Generally speaking, however, the Coulombic singularity of the nuclear attraction and/or electron-electron repulsion interactions has presented nearly insurmountable difficulties-necessitating the use of various unappealing approximations such as "softened" Coulomb potentials, 5 reduced dimensionality of physical space, 23 or approximate 3-dimensional (3D) DVR potentials. 24,25 One noteworthy exception is the recent work of Martinez and co-workers, who have successfully applied tensor product representations of the electron interaction tensor, in the context of traditional electronic structure calculations. ...
Using a combination of ideas, the ground and several excited electronic states of the helium atom and the hydrogen molecule are computed to chemical accuracy—i.e., to within 1–2 mhartree or better. The basic strategy is very different from the standard electronic structure approach in that the full two-electron six-dimensional (6D) problem is tackled directly, rather than starting from a single-electron Hartree-Fock approximation. Electron correlation is thus treated exactly, even though computational requirements remain modest. The method also allows for exact wave functions to be computed, as well as energy levels. From the full-dimensional 6D wave functions computed here, radial distribution functions and radial correlation functions are extracted—as well as a 2D probability density function exhibiting antisymmetry for a single Cartesian component. These calculations support a more recent interpretation of Hund’s rule, which states that the lower energy of the higher spin-multiplicity states is actually due to reduced screening, rather than reduced electron-electron repulsion. Prospects for larger systems and/or electron dynamics applications appear promising.
... They applied a spectral scheme using the rational Legendre functions for solving the Kortewegde Vries equation on the half line. Boyd et al. (2003) have applied pseudospectral methods on a semi-infinite interval and compared the rational Chebyshev, Laguerre and the mapped Fourier Sine methods. Parand et al. (2013 and Parand and Delkhosh (2017) have applied spectral methods such as rational tau and collocation methods to solve nonlinear ordinary differential equations on semiinfinite intervals. ...
In this paper, the combination of quasilinearization and collocation methods is used for solving the problem of the boundary layer flow of Eyring–Powell fluid over a stretching sheet. The proposed approach is based on Hermite function collocation method. The quasilinearization method is used for converting the non-linear Eyring–Powell problem to a sequence of linear equations and the Hermite collocation method is applied for solving linear equations at each iteration. In the end, the obtained result of the present work is compared with the obtained results in other papers.
... They applied a spectral scheme using the rational Legendre functions for solving the Korteweg-de Vries equation on the half-line. Boyd et al. [68] applied pseudo-spectral methods on a semi-infinite interval and compared rational Chebyshev, Laguerre and mapped Fourier sine methods. Parand et al. [30,31,32,33,34,35,63] also applied spectral method to solve nonlinear ordinary differential equations defined over the interval I = [0, ∞). ...
A series of problems in different fields such as physics and chemistry are modeled by differential equations. Differential equations are divided into partial differential equations and ordinary differential equations which can be linear or nonlinear. One approach to solve those kinds of equations is using orthogonal functions into spectral methods. In this paper, we firstly describe Laguerre, Hermite, and Sinc orthogonal functions. Secondly, we select three interesting problems which are modeled as differential equations over the interval $[0, +\infty)$. Then, we use the collocation method as a spectral method for solving those selected problems and compare the performance of Laguerre, Hermite, and Sinc orthogonal functions in solving those types of equations.
... In order to achieve a balanced description for a large range of quantum numbers, and for the sake of higher accuracy, the radial Schrödinger equation is solved on a semi-infinite grid via an expansion in Chebyshev polynomials. 69 ...
Electronic excitations of an electron bound to an alkali metal ion inside a droplet of superfluid ⁴He are computed via a combination of helium density functional theory and the numerical integration of the Schrödinger equation for a single electron in a modified, He density dependent atomic pseudopotential. The application of a spectral method to the radial part of the valence electron wavefunction allows the computation of highly excited Rydberg states. For low principal quantum numbers the energy required to push the electron outward is larger than the solvation energy of the ion. However, for larger principal quantum numbers the situation is reversed, which suggests the stability of a system where the ion sits inside the droplet while the valence electron orbits the nanodroplet.
... In fact, it is shown in [23,24] that the Legendre and Chebyshev rational approximations are an efficient spectral methods for problems in semi-infinite intervals. Boyd et al. [6] proposed spectral collocation methods on a semi-infinite interval and compared rational Chebyshev, Laguerre and mapped Fourier sine. Parand et al. [28−31] Chebyshev and Legendre rational functions with tau and collocation method were applied to solve nonlinear ordinary differential equations on semi-infinite intervals. ...
A new spectral Jacobi rational-Gauss collocation (JRC) method is proposed for solving the multi-pantograph delay differential equations on the half-line. The method is based on Jacobi rational functions and Gauss quadrature integration formula. The main idea for obtaining a semi-analytical solution for these equations is essentially developed by reducing the pantograph equations with their initial conditions to systems of algebraic equations in the unknown expansion coefficients. The convergence analysis of the method is analyzed. The method possesses the spectral accuracy. Numerical results indicating the high accuracy and effectiveness of this algorithm are presented. Indeed, the present method is compared favorably with other methods. © 2017, Institute of Applied Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg.
... The next approach is reformulating the original problem to a singular problem in bounded domain and then using suitable Jacobi approximation to resolve the resulting singular problem (Guo, 2000(Guo, , 2003. Another effective method is based on rational orthogonal functions, such as the rational Legendre functions and rational Chebyshev functions which are mutually orthogonal in semi-infinite intervals (Guo et al., , 2002Shen and Wang, 2009;Boyd et al., 2003). Also we refer the interested reader to Doha et al. (2014a, b, c, d), , Bhrawy and Zaky (2015) and Abbasbandy et al. (2014) for more research works in the solution of differential equations in semi-infinite domains. ...
Purpose
The purpose of this paper is to develop an efficient method for solving the magneto-hydrodynamic (MHD) boundary layer flow of an upper-convected Maxwell (UCM) fluid over a porous isothermal stretching sheet.
Design/methodology/approach
The paper applied a collocation approach based on rational Legendre functions for solving the third-order non-linear boundary value problem, describing the MHD boundary layer flow of an UCM fluid over a porous isothermal stretching sheet. This method solves the problem on the semi-infinite domain without transforming domain of the problem to a finite domain.
Findings
This approach reduces the solution of a problem to the solution of a system of algebraic equations. The numerical values of the skin friction coefficient are presented and analyzed for various parameters of interest in the problem. The authors also compare the results of this work with some recent results and show that the new method is efficient and applicable.
Originality/value
The method solves this problem without use of discrete variables and linearization or small perturbation. Also it was confirmed by the theorem and figure of absolute coefficients that this approach has exponentially convergence rate.
... They applied a spectral scheme using the rational Legendre functions for solving the Korteweg-de Vries equation on the half-line. Boyd et al. [21] applied pseudospectral methods on a semi-infinite interval and compared rational Chebyshev, Laguerre and mapped Fourier sine methods. ...
In this paper, we propose Hermite collocation method for solving Thomas-Fermi equation that is nonlinear ordinary differential equation on semi-infinite interval. This method reduces the solution of this problem to the solution of a system of algebraic equations. We also present the comparison of this work with solution of other methods that shows the present solution is more accurate and faster convergence in this problem.
... Later, Guo et al. [7] developed Boyd's idea and introduced rational Legendre functions. The rational Chebyshev and rational Legendre functions have been successfully used in a wide range of applications [3][4][5][6][7][8][9][10][11][12]. ...
In this paper, a generalization of rational Chebyshev functions and named fractional rational Chebyshev functions, is introduced for solving fractional differential equations. By using the collocation scheme, the efficiency and performance of the new basis is shown through several examples. Also, the obtained results are compared with rational Chebyshev results. It is shown that the generalized functions are more efficient to solve fractional differential equations, and they converge more rapidly. © 2014, Azerbaijan National Academy of Sciences. All rights reserved.
... They applied to a spectral scheme using the rational Legendre functions for solving the Korteweg-de Vries equation on the half-line. Boyd et al. [7] applied pseudospectral methods on a semiinfinite interval and compared this with the rational Chebyshev, Laguerre and mapped Fourier sine methods. ...
A new approach, named the exponential function method (EFM) is used to obtain solutions to nonlinear ordinary differential equations with constant coefficients in a semi-infinite domain. The form of the solutions of these problems is considered to be an expansion of exponential functions with unknown coefficients. The derivative and product operational matrices arising from substituting in the proposed functions convert the solutions of these problems into an iterative method for finding the unknown coefficients. The method is applied to two problems: viscous flow due to a stretching sheet with surface slip and suction; and mageto hydrodynamic (MHD) flow of an incompressible viscous fluid over a stretching sheet. The two resulting solutions are compared against some standard methods which demonstrates the validity and applicability of the new approach.
... Domain truncation was coupled with shooting, finite differences and Galerkin schemes. Galerkin or collocation methods based on the Laguerre functions, and the Chebyshev polynomials with various mappings, are the other two alternatives (see for instance the paper of Boyd et al. [11]). ...
We make use of two Laguerre collocation techniques as a way to illustrate, deepen or extend some analytic results of existence, uniqueness and asymptotic behavior concerning the solutions to second and third orders nonlinear boundary value problems on the half line. We point out some particular features of solutions, e.g. boundary layers, which have not been noticed by theoretical studies and accurately resolve some analytic singularities. With respect to some challenging BVPs from engineering, we show that not absolutely all the analytic results are validated numerically. The free parameter, i.e., the scaling factor, involved in the Laguerre weight function is adjusted in order to ensure the most efficient convergence of the Newton type algorithms.
... They applied a spectral scheme using the rational Legendre functions for solving the Korteweg-de Vries equation on the half-line. Boyd et al. [20] applied pseudospectral methods on a semi-infinite interval and compared rational Chebyshev, Laguerre and mapped Fourier sine methods. ...
In this paper, we decide to compare rational and exponential Legendre functions Tau approach to solve the governing equations for the flow of a third grade fluid in a porous half space. Firstly, we estimate an upper bound for function approximation based on mentioned functions in semi-infinite domain, and discuss that the analytical functions have a superlinear convergence for these basis. Also the operational matrices of derivative and product of these functions are presented to reduce the solution of this problem to the solution of a system of nonlinear algebraic equations. The comparison of the results of rational and exponential Legendre Tau methods with numerical solution shows the efficiency and accuracy of these methods. We also make a comparison between these two methods themselves and show that using exponential functions, leads to more accurate results and faster convergence in this problem.
... Boyd 等人 [168] 则应用半直线上的拟谱方法数值模拟氢原子行为, 同时比较了带变量变换的 Fouriersine 展开方法、Laguerre 展开方法和 Chebyshev 无理基函数展开方法. ...
In this paper, we review the current trends and progresses in spectral and pseudospectral methods for unbounded domains and exterior problems. The rst kind of numerical methods are based on the orthogonal approximations and interpolations by using the Hermite polynomials and functions, and the Laguerre polynomi-fials and functions. The second kind of numerical methods are based on the Jacobi orthogonal approximations and interpolations with suitable variable transformations. The third kind of numerical methods are induced by the various combinations of the previous orthogonal approximations and interpolations coupled with domain decomposition and other techniques. We also present the main results on the Hermite, Laguerre, Jacobi irra-fitional orthogonal approximations and interpolations, which serve as the mathematical foundation of the related numerical algorithms.
... The constant parameter L sets the length scale of the mapping. Boyd offered guidelines for optimizing the map parameter L where L > 0 in [5,39,40]. In general, there is no way to avoid trial and error in choosing L when solving problems on an unbounded domain [41]. ...
In this paper, a numerical method for solving Lane-Emden type equations, which are nonlinear ordinary differential equations on the semi-infinite domain, is presented. The method is based upon the modified rational Bernoulli functions; these functions are first introduced. Operational matrices of derivative and product of modified rational Bernoulli functions are then given and are utilized to reduce the solution of the Lane-Emden type equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. Copyright © 2015 John Wiley & Sons, Ltd.
... In quantum chemistry, Friesner and others have shown orders of magnitude improvement in speed for a wide variety of methods [24][25][26][27][28][29][30][31][32]. Direct solution of Schrödinger's equation has been performed for one-electron problems [33,34], but only recently has a sufficient representation of the computational domain been demonstrated for fully correlated, two-electron atoms [35,36]. We use the implementation of Ref. [36]. ...
Two electrons at the threshold of ionization represent a severe test case for electronic structure theory. Variational methods can yield highly accurate energies, but may be less accurate for other operators of interest. A pseudospectral method yields very accurate expectations values for the two-electron ion with nuclear charge at and close to the critical value which yields zero ionization energy. As an illustration, the ground-state density is calculated and an extremely accurate parametrization is given. Other components in Kohn-Sham density functional theory are also calculated, and the efficacy of approximations is discussed.
... They applied a spectral scheme using the rational Legendre functions for solving the Korteweg-de Vries equation on the half line. Boyd et al. [10] applied pseudospectral methods on a semi-infinite interval and compared the rational Chebyshev, Laguerre, and the mapped Fourier sine methods. Parand et al. [11] compared two common collocation approaches based on radial basis functions for the case of heat transfer equations arising in porous medium. ...
A new shifted Legendre-Gauss collocation method is proposed for the solution of Volterra's model for population growth of a species in a closed system. Volterra's model is a nonlinear integrodifferential equation on a semi-infinite domain, where the integral term represents the effects of toxin. In this method, by choosing a step size, the original problem is replaced with a sequence of initial value problems in subintervals. The obtained initial value problems are then step by step reduced to systems of algebraic equations using collocation. The initial conditions for each step are obtained from the approximated solution at its previous step. It is shown that the accuracy can be improved by either increasing the collocation points or decreasing the step size. The method seems easy to implement and computationally attractive. Numerical findings demonstrate the applicability and high accuracy of the proposed method.
... line. Boyd et al. 17 applied pseudospectral methods on a semi-infinite interval and compared rational Chebyshev, Laguerre and mapped Fourier sine. ...
In this paper unsteady isothermal flow of a gas through a semi-infinite Micro-Nano porous medium which is a non-linear two-point boundary value problem (BVP) on semi-infinite interval has been considered. We solve this problem by two different pseudospectral approaches and compare their results with solution of other methods. The proposed approaches are equipped by the orthogonal rational Legendre and Sinc functions. These methods reduce solution of the problem to solution of a system of algebraic equations. Also through the convergence of these methods we determine the accurate initial slope y'(0) with good capturing the essential behavior of y(x).
... In these schemes, the coordinate mapping method was applied to improve the efficiency of the grid points with sampling denser grid points in the physically more important regions [18][19][20][21]. It has been proven that the mapped FFT method works efficiently with the atomic Coulomb singularity in spherical coordinates [19,[21][22][23]. However, the accuracy of the mapped FFT for the Coulomb singularities in a molecule is not so satisfactory [19]. ...
... Boyd [15] applied pseudo-spectral methods on a semi-infinite interval and compared rational Chebyshev, Laguerre, and mapped Fourier sine method. Moreover, many researchers have investigated (scrutinized) on spectral methods, for example [16 -19]. ...
In this paper, a new numerical algorithm is introduced to solve the Blasius equation, which is a third-order nonlinear ordinary differential equation arising in the problem of two-dimensional steady state laminar viscous flow over a semi-infinite flat plate. The proposed approach is based on the first kind of Bessel functions collocation method. The first kind of Bessel function is an infinite series, defined on R and is convergent for any x is an element of R. In this work, we solve the problem on semi-infinite domain without any domain truncation, variable transformation basis functions or transformation of the domain of the problem to a finite domain. This method reduces the solution of a nonlinear problem to the solution of a system of nonlinear algebraic equations. To illustrate the reliability of this method, we compare the numerical results of the present method with some well-known results in order to show the applicability and efficiency of our method.
... Boyd [12] applied pseudo-spectral methods on a semi-infinite interval and compared Rational Chebyshev, Laguerre and mapped Fourier sine. Moreover, many researchers have investigated (scrutinized) on spectral methods, for example, [13,14,15,16,17,18,19,20,21,22,23,24,25,26] In this paper, we attempt to introduce a new method, based on the modified Bessel functions of the first kind for solving unbounded problems. ...
Volterra's model for population growth in a closed system includes an integral term to indicate accumulated toxicity in addition to the usual terms of the logistic equation. In this research, a new numerical algorithm is introduced for solving this model. The proposed numerical approach is based on the modified Bessel function of the first kind and the collocation method. In this method, we aim to solve the problems on the semi-infinite domain without any domain truncation, variable transformation in basis functions and shifting the problem to a finite domain. Accordingly, we employ two different collocation approaches, one by computing through Volterra's population model in the integro-differential form and the other by computing by converting this model to an ordinary differential form. These methods reduce the solution of a problem to the solution of a nonlinear system of algebraic equations. To illustrate the reliability of these methods, we compare the numerical results of the present methods with some well-known results in other to show that the new methods are efficient and applicable.
The numerical method based on Sinc function is applied for the solution of Rosenau–KdV equation in this paper. The equation is fully-discretized by using the Sinc collocation method for spatial discretization and the forward finite difference for time discretization. The difference scheme is indicated to be conditionally stable using error analysis. The validity and accuracy of this method are verified by serval numerical experiments using single solitary waves, double solitary waves and multiple solitary waves.
Hartree-Fock calculations are performed with quadruple-precision arithmetic for 117 single atoms, from He to Og (Z = 118), using Lambda functions. Lambda functions are typical Laguerre-type basis functions, and are suitable for constructing complete orthonormal system for bound states. Through computation, the performance of Lambda functions are investigated. The number of expansion terms for s-type, p-, d- and f-type symmetries are respectively 150, 149, 148, and 147. The numbers of significant figures of the total energies obtained for Group 18 atoms (He, Ne, Ar, Kr, Xe, Rn, Og) are respectively 30, 30, 30, 23, 20, 17, and 15. For Group 1 atoms (Li, Na, K, Rb, Cs, and Fr), the numbers of significant figures are respectively 30, 28, 22, 17, 14, and 13. High precision results, to 30 digits, can be obtained for atoms from the first to the third period (except Na, Al, Si). For atoms with greater atomic number, however, accuracy was degraded. A much larger expansion is necessary to ensure accuracy when the cusp condition is relevant. Values of the energy, virial ratios, orbital energies, and expectation values of rn (n = 2, 1, −1, −2) are given in the Supplementary Material.
This paper revisits the generalized pseudospectral (GPS) method on the calculation of various radial expectation values of atomic systems, especially on the spatially confined hydrogen atom and harmonic oscillator. As one of the collocation methods based on global functions, the powerfulness and robustness of the GPS method has been well established in solving the radial Schrödinger equation with high accuracy. However, in our recent work, it was found that the previous calculations based on the GPS method for the radial expectation values of confined systems show significant discrepancies with other theoretical methods. In this work we have tackled such a problem by tracing its source to the GPS method and found that the method itself may not be able to obtain the system wave function at the origin. Combined with an extrapolation method developed here, the GPS method can fully reproduce the radial quantities obtained by other theoretical methods, but with more flexibility, efficiency, and accuracy. We apply the GPS‐extrapolation method to investigate the relatievistic fine structure and hyperfine splitting of confined hydrogen atom in s‐wave states where the zero‐point wave function dominates. Good agreement with previous predictions is obtained for confined hydrogen in low‐lying states, and benchmark results are obtained for high‐lying excited states. The perturbation treatment of the fine and hyperfine interactions is validated in the confining environment. The generalized pseudospectral (GPS) method has demonstrated its powerfulness in solving the radial Schrödinger equation by approximating the unknown function through a superposition of delta‐like functions, with coefficients being exactly the values of unknown function. In this work, we solve the undiscovered problem where the method itself cannot produce the wave function at origin. An extrapolation procedure must be combined with the GPS method to calculate various physical quantities.
In this research, we present an iterative spectral method for the approximate solution of a class of Lane–Emden equations. In this procedure, we initially extend the Legendre wavelet which is appropriate for any time interval. Thereafter, the Guass-Legendre collection points of the Legendre wavelet are acquired. Employing this new approach, the iterative spectral technique converts the differential equation to a set of algebraic equations which diminishes the computational costs effectively. By solving the obtained algebraic equations, an accurate approximate solution for the assumed Lane–Emden equation is achieved. The present technique is validated by solving a number of Lane–Emden problems and are compared with other existing methods. The numerical simulations demonstrate that the new algorithm is simple and it has highly accuracy.
In this paper, an iterative method is introduced for the numerical solution of a class of nonlinear two-point boundary value problems (BVPs) on semi-infinite intervals. The underlying strategy behind this novel approach is to construct a tailored integral operator that is expressed in terms of a Green's function for the corresponding linear differential operator of the BVP. Then, two well-known fixed point iterations, including Picard's and Krasnoselskii-Mann's schemes, are applied to this integral operator that results in this new iterative technique. A proof of convergence of the numerical scheme, based on the contraction principle, is included. We demonstrate the reliability, fast convergence, applicability of the method and compare its performance, using some relevant test examples that appear in the literature.
In this study, we propose an efficient and accurate numerical technique that is called the rational Chebyshev collocation (RCC) method to solve the two dimensional flow of a viscous fluid in the vicinity of a stagnation point named Hiemenz flow. The Navier-Stokes equations governing the flow, are reduced to a third-order ordinary differential equation of a boundary value problem with a semi-infinite domain by using similarity transformation. The rational Chebyshev method reduces this nonlinear ordinary differential equation to a system of algebraic equations. This technique is a powerful type of the collocation methods for solving the boundary value problems over a semi-infinite interval without truncating it to a finite domain. We also present the comparison of this work with others and show that the present method is more accurate and efficient.
In finite difference methods the exact solution v of the differential equation is approximated by low-order polynomials interpolating v at several nearby mesh points. For example, ( 9.3) is an approximation of the derivative at \(x_j\) obtained by parabolic interpolation between \(x_{j-1}\), \(x_j\), and \(x_{j+1}\). The solution on the whole interval is constructed by superposing many such overlapping polynomials as the weighted sum of the function values at the interpolation points.
We present the ORTHOPOLY software that permits to evaluate, efficiently and accurately, finite series of any classical family of orthogonal polynomials (Chebyshev, Legendre, ultraspherical or Gegenbauer, Jacobi, Hermite and Laguerre orthogonal polynomials) and their derivatives. The basic algorithm is the BCS-algorithm (Barrio–Clenshaw–Smith derivative algorithm), that permits to evaluate the kth derivative of a finite series of orthogonal polynomials at any point without obtaining before the previous derivatives. Due to the presence of rounding errors, specially in the case of high order derivatives, we introduce the compensated BCS-algorithm, based on Error-Free Transformation techniques, that permits to relegate the influence of the conditioning of the problem up to second order in the round-off unit of the computer. The BCS and compensated BCS algorithms may also give running-error bounds to provide information about the accuracy of the evaluation process. The ORTHOPOLY software includes C and Matlab versions of all the algorithms, and they are designed to be easily used in longer softwares to solve physical, mathematical, chemical or engineering problems (illustrated on the Schrödinger equation for the radial hydrogen atom).
Program summary
Program Title: ORTHOPOLY
Program Files doi: http://dx.doi.org/10.17632/n55bpy5bsr.1
Licensing provisions: GPLv3
Programming language: C and Matlab versions
Nature of problem: Accurate numerical evaluation of finite series of classical orthogonal polynomials and their derivatives.
Solution method:Barrio–Clenshaw–Smith algorithm for the evaluation of derivatives of finite series of classical orthogonal polynomials. Error-Free Transformation techniques for the Compensated Barrio–Clenshaw–Smith algorithm in order to provide accurate evaluations. Running-error techniques to provide error bounds of the evaluations.
Purpose
In this paper a new numerical method is proposed for the solution of the Blasius and magnetohydrodynamic (MHD) Falkner-Skan boundary-layer equations. The Blasius and MHD Falkner-Skan equations are third order nonlinear boundary value problems on the semi-infinite domain.
Design/methodology/approach
The approach is based upon modified rational Bernoulli functions. The operational matrices of derivative and product of modified rational Bernoulli functions are presented. These matrices together with the collocation method are then utilized to reduce the solution of the Blasius and MHD Falkner-Skan boundary-layer equations to the solution of a system of algebraic equations.
Findings
The method is computationally very attractive and gives very accurate results.
Originality/value
Many problems in science and engineering are set in unbounded domains. One approach to solve these problems is based on rational functions. In this work, a new rational function is used to find solutions of the Blasius and MHD Falkner-Skan boundary-layer equations.
The representation of spatial derivatives is at the heart of spectral methods for partial differential equations, so the three main kinds (Fourier, Chebyshev and Legendre) are analyzed at the outset, together with efficient means to compute them. Galerkin methods involving all three classes of basis functions are discussed for both stationary (Helmholtz equation) and non-stationary (advection equation) problems. Tau methods are also applicable to both types of problems and are presented next. Due to their straightforward implementation of boundary conditions, they offer an exciting alternative to Galerkin approaches. Separate Sections are devoted to collocation methods and efficient means to solve non-linear equations in the spectral framework. Useful hints for time integration and dealing with semi-infinite and infinite definition domains are given. The Examples and Problems include the Galerkin method for the advection, diffusion and Poisson equations (Poiseuille flow), the tau method for the Poisson equation, the diffusion equation in collocation approaches, and the Burgers equation.
Spectral and pseudospectral methods in chemistry and physics are based on classical and nonclassical orthogonal polynomials defined in terms of a three term recurrence relation. The coefficients in the three term recurrence relations for the nonclassical polynomials can be calculated with the Gautschi-Stieltjes procedure. The round-off errors that occur with the use of Gram-Schmidt orthogonalization procedure is demonstrated for both classical and nonclassical polynomials. The trapezoidal, Simpson’s and Newton-Cotes integration rules are derived as are the Fejér, Clenshaw-Curtiss, Gauss-Lobatto and Gauss-Radau algorithms. Sinc interpolation based on Fourier sine basis functions is compared with the Lagrange interpolation. Nonclassical Maxwell and Bimodal polynomials orthogonal on the infinite domain with respect to weight functions \(w(x) = x^2\exp (-x^2)\) and \(x^2\exp [-(x^4/4\epsilon -x^2/2\epsilon )]\), respectively, are introduced for kinetic theory problems. The Gaussian quadrature rule based on the nonclassical Rys polynomials orthogonal with respect to the weight function \(w(x) = e^{-cx^2},\; x \in [-1,\ 1]\), used to evaluate integrals in molecular quantum mechanics is presented. For \(c \rightarrow 0\) and \(c \rightarrow \infty \), the Rys polynomials are the Legendre and scaled Hermite polynomials, respectively. Two dimensional quadratures, such as the Lebedev cubature, are used to evaluate two dimensional integrals in density functional theory for electronic structure calculations as well as for the nonlinear Boltzmann equation in kinetic theory. The Stieltjes moment problem is related to the inversion of moment data in chemical physics to reconstruct photoelectron cross sections.
This chapter introduces the basic principles of spectral/pseudospectral methods for the solution of partial differential and/or integral equations that serve to model a large number of physical processes in chemistry and physics. The first part of the chapter defines the spectral space representation of functions and the transformation to the physical space representation. A Hilbert space is defined as well as the definition of self-adjoint operators that occur in quantum mechanics and kinetic theory. The Rayleigh-Ritz variational principle and the method of weighted residuals are discussed. An historical summary of the development of pseudospectral methods in chemistry and physics is presented together with an outline of the book. The science, the mathematical models and the computer algorithms are interrelated.
The orthogonal basis sets most often used in spectral methods are the Chebyshev and Legendre polynomials on a bounded domain, or a Fourier basis set for periodic functions. We discuss in this chapter the expansions of Gaussian and Kappa distributions of kinetic theory in Hermite and Laguerre polynomials on the infinite and semi-infinite intervals, respectively. The spectral convergence properties of these expansions is demonstrated numerically and analytically. The expansions of \(\sin (x)\) in Hermite polynomials, and of the Maxwellian distribution in Chebyshev polynomials are also considered. The basic principles of Fourier series are presented and applied to quantum mechanical wave packets as well as the analysis of free induction decay signals. The resolution of the Gibbs phenomenon with the Gegenbauer reconstruction method is compared with the inverse polynomial reconstruction method. A resolution of the Runge phenonmena is also presented.
The chapter is devoted to the efficient implementation of Chebyshev collocation method. First, the performances of the method in solving fourth order GEPs are compared with those of ChT and ChG counterparts. Then, ChC method is used to solve some genuinely high order, i.e., larger than two, and/or singularly perturbed eigenvalue problems. Two of them, of sixth and eighth order represent linear hydrodynamic stability problems. Also some fourth order problems with variable coefficients (tensile instabilities of thin annular plates etc.) are successfully considered. In order to reduce the high order problems to systems of second order equations supplied with Dirichlet boundary conditions we introduce a so called “\(D^{(2)}\)” strategy or factorization. Using this strategy with \(N=2^{10}\) a conjecture with respect to the first eigenvalue of the Viola’s problem is stated. This is a fourth order singularly perturbed eigenvalue problem. A special attention is paid to the well known Mathieu’s system as a MEP. A lot of eigenmodes and eigenfrequencies corresponding to various geometries of the vibrating elliptic membrane problem, in which this system is originated, are displayed. In order to avoid spurious eigenvalues (at infinity) and to improve the computation of a specified region of the spectrum, mainly in case of large problems, some Jacobi Davidson type methods are used. Making use of the pseudospectrum of a singular GEP we comment on the backward stability and the order of convergence of JD and Arnoldi methods in computing the first two eigenvalues.
The present study is an attempt to find a solution for steady flow of a third-grade fluid by utilizing spectral methods based on rational Christov functions. This problem is described as a nonlinear two-point boundary value problem. The following method tries to solve the problem on the infinite domain without truncating it to a finite domain and transforms the domain of the problem to a finite domain. Researchers in this try to solve the problem by using anew modified rational Christov functions and normal rational Christov function. Finally, the findings of the current study, i.e., proposal methods, numerical out cames and other methods were compared with each other.
Efficient numerical solver for the Schrödinger equation is very important in physics and chemistry. The finite element discrete variable representation (FE‐DVR) was first proposed by Rescigno and Mc‐Curdy [Phys. Rev. A 62, 032706 (2000)] for solving quantum‐mechanical scattering problems. In this work, an FE‐DVR method in a mapped coordinate was proposed to improve the efficiency of the original FE‐DVR method. For numerical demonstration, the proposed approach is applied for solving the electronic eigenfunctions and eigenvalues of the hydrogen atom and vibrational states of the electronic state 3Σg + of the Cs2 molecule which has long‐range interaction potential. The numerical results indicate that the numerical efficiency of the original FE‐DVR has been improved much using our proposed mapped coordinate scheme.
The Lagrange-mesh method is an approximate variational method taking the form of equations on a grid thanks to the use of a Gauss-quadrature approximation. The variational basis related to this Gauss quadrature is composed of Lagrange functions which are infinitely differentiable functions vanishing at all mesh points but one. This method is quite simple to use and, more importantly, can be very accurate with small number of mesh points for a number of problems. The accuracy may however be destroyed by singularities of the potential term. This difficulty can often be overcome by a regularization of the Lagrange functions which does not affect the simplicity and accuracy of the method.
The problem of the boundary layer flow of an incompressible viscous fluid over a non-linear stretching sheet is considered. A spectral collocation method is performed in order to find an analytical solution of the governing nonlinear differential equations. The obtained results are finally compared through the illustrative graphs and tables with the exact solution and some well-known results obtained by other researchers. The comparison shows that the obtained results with the rational Chebyshev collocation method are more accurate.
The efficiency of the time propagators is very important for solving the reactive scattering Schrödinger equation in a time-dependent wave packet (TDWP) calculation. In this work, an efficient 4th order split operators (HOSO), which was presented in the work by Blanes and Moan and was a partitioned Runge-Kutta type integrator (J. Comput. Appl. Math. 142, 313 (2002)), is recommended for a general usage in a triatomic reactive scattering calculation by the TDWP method. This HOSO is constructed in a TVT form, and is an optimal one among a series of HOSO after examining with several typical triatomic reactive scattering processes, H + H2, H + H2+, H + NH, H + O2 and F + HD reactions. A detailed comparison between the performance of HOSO in the VTV form, which was reported by Sun et al. (Phys. Chem. Chem. Phys. 14, 1827 (2012)), and in the TVT form reported in the current work, suggests that this HOSO in the TVT form always have good numerical efficiency. This fact may suggest that this 4th-order propagator in the TVT form can be safely chosen without any further examination, at least among all of the HOSO tested in this work, to apply in an efficient time-dependent wave packet numerical calculation for describing a triatomic reactive scattering process.
The representation of a quantum system by an evenly spaced Fourier grid is examined. This grid faithfully represents wave functions whose projection is contained in a rectangular phase space. This is mathematically equivalent to a band limited function with finite support. In general, wave packets decay exponentially in classically forbidden regions of phase space. This idea is then used first to optimize the rectangular shape of the Fourier grid, leading to exponential convergence. Nevertheless, in most cases the representation is suboptimal. The representation efficiency can then be extremely enhanced by mapping the coordinates. The mapping procedure reshapes the wave function to fit into the rectangular Fourier shape such that the wasted phase space area is minimal. It is shown that canonical transformations, which rescale the coordinates, improve the representation dramatically. A specific scaling transformation enables the representation of the notoriously difficult Coulomb potentials. The scaling transformation enables one to extract almost as many converged eigenstate energies as there are grid points. The method is extendible to more than one dimension, which is demonstrated by the study of the H2+ problem. This scaling transformation can bridge the gap between quantum chemistry and quantum molecular dynamics by enabling the treatment of electronic problems in the vicinity of Coulomb potentials by grid methods developed for molecular dynamics. © 1996 The American Physical Society.
A discussion of the discrete ordinate method for solving differential equations is presented along with a number of examples that have application in various fields of physics. In particular, diffusion cooling, boundary layer meteorology and the diffusion of water in soils are studied. It is shown that the discrete ordinate method is considerably more accurate than finite difference methods of the same order. Results are presented for linear and nonlinear models, with a comprehensive analysis of the results and accuracies.
It is shown that spectral and pseudospectral methods can lead to great savings in numerical efficiency-typically, at least a factor of four in storage and eight in operation count-in comparison with finite-difference methods for solving eigenvalue and nonseparable, noniterative boundary value problems at the cost of only a a slight amount of additional programming. The special problems or apparent problems that geophysical boundary layers, critical latitudes and critical levels and real data create for spectral algorithms are also discussed along with when and how these can be solved.
A theoretical analysis of the effect of a longshore mean shear flow on edge waves is performed in the framework of linear shallow water equations. A single equation describing edge waves as well as neutral shear waves is obtained. A numerical method of calculation is given for the dispersion relations and the wave pattern, accounting for any beach topography and any mean flow profile (remaining constant alongshore and with a straight shoreline). Numerical calculations are presented for a simple exponential flow profile and for a plane bottom. A Doppler shift in the frequencies and a variation in the offshore extension of the waves are found, depending on the maximum local Froude number of the current, F, defined as F = [V(x)/&sqrt;gH(x)]max, where V(x) stands for the mean longshore current, H(x) for the depth, and g for the gravity. The maximum shift in frequency is for wave numbers of about Vx(0)2/gm and frequencies of about Vx(0), where Vx(0) is the shear at the shoreline and m is the beach slope. For instance, these maximum differences may reach about 40% for F=0.5. Waves of any wavelength can always propagate downstream, but they can propagate upstream only for F≤Fc∼0.7. For mean flows with F≥Fc only waves shorter or longer than some forbidden wavelengths can propagate against the current. An analytical dispersion relation of asymptotic general validity for short waves (corresponding to the gravity range in real beaches) is given. The numerical model as well as this analytical dispersion relation is tested by means of a nonplanar real topography.
When used in the on-the-grid solvers of the stationary or time-dependent Schrödinger equation, coordinate mapping allows one to achieve a very accurate description of the wave function with an optimal number of the grid points. The efficiency of the mapped Fourier grid methods has been recently demonstrated by V. Kokoouline, O. Dulieu, R. Kosloff, and F. Masnou-Seeuws [J. Chem. Phys. 110, 9865 (1999)] and by D. Lemoine [Chem. Phys. Lett. 320, 492 (2000)]. In this paper we propose a discrete coordinate representation based on a numerical mapping in cylindrical and spherical coordinates. Within proposed approach, the Hamiltonian matrix is Hermitian, and the use of the fast cosine and sine Fourier transforms provides a very efficient way of calculating the Laplacian operator. © 2001 American Institute of Physics.
“Domain truncation” is the simple strategy of solving problems onyε [-∞, ∞] by using a large but finite computational interval, [− L, L] Sinceu(y) is not a periodic function, spectral methods have usually employed a basis of Chebyshev polynomials,T
n(y/L). In this note, we show that becauseu(±L) must be very, very small if domain truncation is to succeed, it is always more efficient to apply a Fourier expansion instead. Roughly speaking, it requires about 100 Chebyshev polynomials to achieve the same accuracy as 64 Fourier terms. The Fourier expansion of a rapidly decaying but nonperiodic function on a large interval is also a dramatic illustration of the care that is necessary in applying asymptotic coefficient analysis. The behavior of the Fourier coefficients in the limitn→∞ for fixed intervalL isnever relevant or significant in this application.
Contents PREFACE x Acknowledgments xiv Errata and Extended-Bibliography xvi 1 Introduction 1 1.1 Series expansions .................................. 1 1.2 First Example .................................... 2 1.3 Comparison with finite element methods .................... 4 1.4 Comparisons with Finite Differences ....................... 6 1.5 Parallel Computers ................................. 9 1.6 Choice of basis functions .............................. 9 1.7 Boundary conditions ................................ 10 1.8 Non-Interpolating and Pseudospectral ...................... 12 1.9 Nonlinearity ..................................... 13 1.10 Time-dependent problems ............................. 15 1.11 FAQ: Frequently Asked Questions ........................ 16 1.12 The Chrysalis .................................... 17 2 Chebyshev & Fourier Series 19 2.1 Introduction ..................................... 19 2.2 Fourier series ...........
We make several observations about eigenvalue problems using, as examples, Laplace's tidal equations and the differential equation satisfied by the associated Legendre functions. Whatever the discretization, only some of the eigenvalues of the N-dimensional matrix eigenvalue problem will be good approximations to those of the differential equation-usually the N/2 eigenvalues of smallest magnitude. For the tidal problem, however, the ''good'' eigenvalues are scattered, so our first point is: It is important to plot the ''drift'' of eigenvalues with changes in resolution. We suggest plotting the difference between a low resolution eigenvalue and the nearest high resolution eigenvalue, divided by the magnitude of the eigenvalue or the intermodal separation, whichever is smaller. Second, as a final safeguard, it is important to look at the Chebyshev coefficients of the mode: We show a numerically computed ''anti-Kelvin'' wave which has little eigenvalue drift, but is completely spurious as is obvious from its spectral series, Third, inverting the roles of parameters can drastically modify the spectrum; Legendre's equation may have either an infinite number of discrete modes or only a handful, depending on which parameter is the eigenvalue. Fourth, when the modes are singular but decay to zero at the endpoints (as is true of tides), a tanh-mapping can retrieve the usual exponential accuracy of spectral methods. Fifth, the pseudospectral method is more reliable than deriving a banded Galerkin matrix by means of recurrence relations; the pseudospectral code is simple to check, whereas it is easy to make an untestable mistake with the intricate algebra required for the Galerkin method. Sixth, we offer a brief cautionary tale about overlooked modes, All these cautions are applicable to all forms of spatial discretization including finite difference and finite element methods. However, we limit our illustrations to spectral schemes, where these difficulties are most easily resolved. With a bit of care, the pseudospectral method is a very robust and efficient method for solving differential eigenproblems, even with singular eigenmodes. (C) 1996 Academic Press, Inc.
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Accurate eigenvalues and eigenfunctions of the anharmonic oscillator
(H=p2+x2+λ x4,λ >0)
and the quartic oscillator (H=p2+x4) are obtained
in all regimes of the quantum number n and the anharmonicity constant
λ . Transition moments of comparable accuracy are obtained for
the quartic oscillator. The method, applicable quite generally for
eigenvalue problems, is non-perturbative and involves the use of an
appropriately scaled basis for the determination of each eigenvalue. The
appropriate scaling formula for a given regime of (n,λ) is
constructed from the oscillation properties of the eigenfunctions. More
general anharmonic oscillators are also discussed.
Mapping methods for improving the accuracy of Fourier pseudospectral differentiation on the infinite line are considered. For a certain class of functions, optimal mapping parameters are obtained by balancing the error due to boundary effects with the internal resolution error. Estimates for the maximal rate of convergence yielded by each mapping are also derived. In the case of evolution equations, the effect of these mappings on the stability of the time integration is investigated. The robustness of the optimal mapping parameters with respect to changes in the solution during the time evolution is examined numerically.
Although Hermite functions were widely used for many practical problems, numerical experiments with the standard (normalized) Hermite functions $\psi _n (v)$ worked poorly in the sense that too many Hermite functions are required to solve differential equations. In order to obtain accurate numerical solutions, it is necessary to choose a scaling factor $\alpha $ and use $\psi _n (\alpha v)$ as the basis functions. In this paper the scaling factors are given for functions that are of Gaussian type, which have finite supports $[ - M,M]$. The scaling factor used is $\max _{0 \leqq j \leqq N} \{ \gamma _j \} / M$, where $\{ \gamma _j \} _{j = 0}^N $ are the roots of $\psi _{N + 1} (v)$ and $N + 1$ is the number of the truncated terms used. The numerical results show that after using this scaling factor, only reasonable numbers of the Hermite functions are required to solve differential equations.
Coordinate mapping can be used to dramatically enhance the efficiency of phase space sampling. A new mapped pseudospectral approach that readily exploits a fast transform algorithm is presented for both cylindrical and spherical coordinates. The combined use of a power law mapping and of a sine Fourier expansion requires only two fast transforms per Laplacian operation. The mapping algorithm can be symmetrized for the purpose of efficient diagonalization techniques. Special emphasis is given to the treatment of radial singularities as the new procedure is illustrated with the eigenvalue calculation of the harmonic oscillator, the Coulomb and the H2+ benchmark problems.
The eigenvalue problem of the general anharmonic oscillator (Hamiltonian H2mu(k,lambda)=- d2/dx2+kx2+lambda x2mu,(k,lambda)>0, mu =2,3,4,...) is investigated in this work. Very accurate eigenvalues are obtained in all regimes of the quantum number n and the anharmonicity constant lambda . The eigenvalues, as functions of lambda , exhibit crossings. The qualitative features of the actual crossing pattern are substantially reproduced in the W.K.B. approximation. Successive moments of any transition between two general anharmonic oscillator eigenstates satisfy exactly a linear recurrence relation. The asymptotic behaviour of this recursion and its consequences are examined.
We make several observations about eigenvalue problems using, as examples, Laplace's tidal equations and the differential equation satisfied by the associated Legendre functions. Whatever the discretization, only some of the eigenvalues of theN-dimensional matrix eigenvalue problem will be good approximations to those of the differential equation—usuallytheN/2 eigenvalues of smallest magnitude. For the tidal problem, however, the “good” eigenvalues are scattered, so our first point is: It is important to plot the “drift” of eigenvalues with changes in resolution. We suggest plotting the difference between a low resolution eigenvalue and the nearest high resolution eigenvalue, divided by the magnitude of the eigenvalue or the intermodal separation, whichever is smaller. Second, as a final safeguard, it is important to look at the Chebyshev coefficients of the mode: We show a numerically computed “anti-Kelvin” wave which has little eigenvalue drift, but is completely spurious as is obvious from its spectral series. Third, inverting the roles of parameters can drastically modify the spectrum; Legendre's equation may have either an infinite number of discrete modes or only a handful, depending on which parameter is the eigenvalue. Fourth, when the modes are singular but decay to zero at the endpoints (as is true of tides), a tanh-mapping can retrieve the usual exponential accuracy of spectral methods. Fifth, the pseudospectral method is more reliable than deriving a banded Galerkin matrix by means of recurrence relations; the pseudospectral code is simple to check, whereas it is easy to make an untestable mistake with the intricate algebra required for the Galerkin method. Sixth, we offer a brief cautionary tale about overlooked modes. All these cautions are applicable toallforms of spatial discretization including finite difference and finite element methods. However, we limit our illustrations to spectral schemes, where these difficulties are most easily resolved. With a bit of care, the pseudospectral method is a very robust and efficient method for solving differential eigenproblems, even with singular eigenmodes.
We consider spectral methods for solving differential equations on unbounded domains. In particular, we consider a spectral rational function method and a method based on domain truncation combined with Fourier series. Both methods contain a free parameter which determines the accuracy of the spectral method. When solving evolution equations the optimal parameter may change with time resulting in a significant loss of accuracy if the parameter is kept fixed. We develop an automatic algorithm which adapts the parameter at regular time levels to maintain optimum accuracy. In numerical tests we found that the proposed method is robust and efficient.
A scheme for the systematic improvement of the mapping mechanism of the mapped Fourier method (MFM) is presented. Older schemes, which relied on the WKB approximation to build the adapted grid placed very few points close to the classical turning points of a potential, which deteriorates the accuracy of the sampling of the wave functions that are represented on the grid. A numerical example at the end of the Letter will prove the efficiency of the new scheme.
The effect of Ekman friction on baroclinic instability is reexamined in order to address questions raised by Farrell concerning the existence of normal mode instability in the atmosphere. As the degree of meridional confinement is central to the result, a linearized two-dimensional (latitude-height) quasi-geostrophic model is used to obviate the arbitrariness inherent in choosing a channel width in one-dimensional (vertical shear only) models. The two-dimensional eigenvalue problem was solved by pseudospectral method using rational Chebyshev expansions in both vertical and meridional directions. It is concluded that the instability can be eliminated only by the combination of strong Ekman friction with weak large-scale wind shear. Estimates of Ekman friction based on a realistic boundary-layer model indicate that such conditions can prevail over land when the boundary layer is neutrally stratified. For values of Ekman friction appropriate to the open ocean, friction can reduce the growth rate of ...
By using the map y = L cot(t) where L is a constant, differential equations on the interval yϵ [− ∞, ∞] can be transformed into tϵ [0, π] and solved by an ordinary Fourier series. In this article, earlier work by Grosch and Orszag (J. Comput. Phys.25, 273 (1977)), Cain, Ferziger, and Reynolds (J. Comput. Phys.56, 272 (1984)), and Boyd (J. Comput. Phys.25, 43 (1982); 57, 454 (1985); SIAM J. Numer. Anal. (1987)) is extended in several ways. First, the series of orthogonal rational functions converge on the exterior of bipolar coordinate surfaces in the complex y-plane. Second, Galerkin's method will convert differential equations with polynomial or rational coefficients into banded matrix problems. Third, with orthogonal rational functions it is possible to obtain exponential convergence even for u(y) that asymptote to a constant although this behavior would wreck alternatives such as Hermite or sinc expansions. Fourth, boundary conditions are usually “natural” rather than “essential” in the sense that the singularities of the differential equation will force the numerical solution to have the correct behavior at infinity even if no constraints are imposed on the basis functions. Fifth, mapping a finite interval to an infinite one and then applying the rational Chebyshev functions gives an exponentially convergent method for functions with bounded endpoint singularities. These concepts are illustrated by five numerical examples.
By using the method of steepest descent, I have compared the suitability of three different methods for solving problems in a semi-infinite or infinite domain using Chebyshev polynomials. Exponential mappings are uniformyly bad. Domain truncation and algebraic mapping both work well, but each is superior for a different category of problems. When the solution is an entire function, then domain truncation is best. It is always simpler to apply than algebraic mapping and always at least as accurate—much more accurate for functions which decay very rapidly. For singular functions, on the other hand, algebraic mapping is better because it is less sensitive to the mapping scale factor L, permitting a better compromise in resolving both the singularity and the exponential decay. For both types of problems, I give simple, explicit estimates of both the optimum choise of domain size or mapping factor and of the attainable accuracy. For the model entire function exp [−Azk] on a semi-infinite interval [0, ∞], the optimum domain size is and the smallest attainable error is roughly e−n(0.896)k where n is the number of Chebyshev polynomals used. Similar formulas for singular functions handled via algebraic mapping are given in (3.41) to (3.45) below.
By applying a mapping to the Chebyshev polynomials, we define a new spectral basis: the “rational Chebyshev functions on the semi-infinite interval,” denoted by TLn(y). Continuing earlier work by the author and by Grosch and Orszag, we show that these rational functions inherit most of the good numerical characteristics of the Chebyshev polynomials: orthogonality, completeness, exponential or “infinite order” convergence, matrix sparsity for equations with polynomial coefficients, and simplicity. Seven numerical examples illustrate their versatility. The “Charney” stability problem of meteorology, for example, is solved to show the feasibility of applying spectral methods to a semi-infinite atmosphere. For functions that are singular at both endpoints, such as K1(y), one may combine rational Chebyshev functions with a preliminary mapping to obtain a single, exponentially convergent expansion on yϵ [0, ∞ ]. Finally, we successfully generalize the WKB method to obtain, for the J0 Bessel function, an amplitude-phase approximation which is convergent rather than asymptotic and is accurate not merely for large y but for all y, even the origin.
When a function is singular but infinitely differentiable at the origin, its power series diverges factorially and its Chebyshev coefficients are proportional to exp(-constant nr) for 0 < r < 1. The two case studies presented here are novel by exemplifying the limits r → 0+ and r → 1−, respectively.
This established text and reference contains an advanced presentation of quantum mechanics adapted to the requirements of modern atomic physics. It includes topics of great current interest such as semiclassical theory, chaos and Bose-Einstein condensation in atomic gases. The third edition of Theoretical Atomic Physics extends the successful second edition with a detailed treatment of the wave motion of atoms near the anticlassical or extreme quantum regime, and it also contains an introduction to some aspects of atom optics that are relevant for current and future experiments involving ultra-cold atoms. Various problems are included together with complete solutions. Because it has more emphasis on theory, this book enables the reader to appreciate the fundamental assumptions underlying standard theoretical constructs and to embark on independent research projects.
We analytically compute the asymptotic Fourier coefficients for several classes of functions to answer two questions. The numerical question is to explain the success of the Weideman-Cloot algorithm for solving differential equations on an infinite interval. Their method combines Fourier expansion with a change-of-coordinate using the hyperbolic sine function. The sinh-mapping transforms a simple function like exp(-z2) into an entire function of infinite order. This raises the second, analytical question: What is the Fourier rate of convergence for entire functions of an infinite order? The answer is: Sometimes even slower than a geometric series. In this case, the Fourier series converge only on the real axis even when the function u (z) being expanded is free of singularities except at infinity. Earlier analysis ignored stationary point contributions to the asymptotic Fourier coefficients when u(z) had singularities off the real z-axis, but we show that sometimes these stationary point terms are more important than residues at the poles of u(z). Peer Reviewed http://deepblue.lib.umich.edu/bitstream/2027.42/31817/1/0000763.pdf
A software suite consisting of 17 MATLAB functions for solving differential equations by the spectral collocation (i.e., pseudospectral) method is presented. It includes functions for computing derivatives of arbitrary order corresponding to Chebyshev, Hermite, Laguerre, Fourier, and sine interpolants. Auxiliary functions are included for incorporating boundary conditions, performing interpolation using barycentric formulas, and computing roots of orthogonal polynomials. It is demonstrated how to use the package for solving eigenvalue, boundary value, and initial value problems arising in the fields of special functions, quantum mechanics, nonlinear waves, and hydrodynamic stability.
- J N Bardsley
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Hermite function solution of quantum anharmonic oscillator
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