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Numerical Solution of Isospectral Flows
Abstract
. In this paper we are concerned with the problem of solving numerically isospectral flows. These flows are characterized by the differential equation L 0 = [B(L);L]; L(0) = L 0 ; where L 0 is a d Theta d symmetric matrix, B(L) is a skew-symmetric matrix function of L and [B; L] is the Lie bracket operator. We show that standard Runge--Kutta schemes fail in recovering the main qualitative feature of these flows, that is isospectrality, since they cannot recover arbitrary cubic conservation laws. This failure motivates us to introduce an alternative approach and establish a framework for generation of isospectral methods of arbitrarily high order. 1. Background and notation 1.1. Introduction. The interest in solving isospectral flows is motivated by their relevance in a wide range of applications, from molecular dynamics to micromagnetics to linear algebra. The general form of an isospectral flow is the differential equation L 0 = [B(L); L]; L(0) = L 0 ; (1) where L 0 is a give...
... where A, B(A) are matrices and the spectrum of A is preserved [9,IV.3.2]. See [6] for a detailed discussion of the properties of these systems. A classical example for an isospectral system is the ideal rigid body in its reduced formulation [12], as described by Euler's equations. ...
... Isospectral numerical time integrators were also considered by Diele, Lopez and Politi who proposed the use of the Cayley transform [7]. Calvo, Iserles and Zanna [6] showed that Runge-Kutta methods applied to Eq. 1.1 break isospectrality. To overcome this, they introduce modified Gauss-Legendre Runge-Kutta methods that provide isospectral time integration schemes of arbitrary high order. ...
... To overcome this, they introduce modified Gauss-Legendre Runge-Kutta methods that provide isospectral time integration schemes of arbitrary high order. In [3], Bogfjellmo and Marthinsen proposed higher order symplectic integrators on the cotangent bundle T * G. Modin and Viviani [14] recently developed an alternative route to overcome the obstruction from [6] by applying a Runge-Kutta scheme to the cotangent bundle T * G of a quadratic matrix Lie group G. Reduction to the dual Lie algebra g * with the momentum map was then used to obtain an isospectral integrator, analogous to how Lie-Poisson reduction works in the continuous case to obtain reduced dynamical equations, cf. [12,Ch. ...
Isospectral flows appear in a variety of applications, e.g. the Toda lattice in solid state physics or in discrete models for two-dimensional hydrodynamics. The isospectral property thereby often corresponds to mathematically or physically important conservation laws. The most prominent feature of these systems, i.e. the conservation of the eigenvalues of the matrix state variable, should therefore be retained when discretizing these systems. Recently, it was shown how isospectral Runge-Kutta methods can in the Lie-Poisson case be obtained through Hamiltonian reduction of symplectic Runge-Kutta methods on the cotangent bundle of a Lie group. We provide the Lagrangian analogue and, in the case of symplectic diagonally implicit Runge-Kutta methods, derive the methods through a discrete Euler-Poincaré reduction. Our derivation relies on a formulation of diagonally implicit isospectral Runge-Kutta methods in terms of the Cayley transform, generalizing earlier work that showed this for the implicit midpoint rule. Our work is also a generalization of earlier variational Lie group integrators that, interestingly, appear when these are interpreted as update equations for intermediate time points. From a practical point of view, our results allow for a simple implementation of higher order isospectral methods and we demonstrate this with numerical experiments where both the isospectral property and energy are conserved to high accuracy.
... Differential equations evolving on the space of symmetric positive definite matrices occur in different contexts. One example is in the inverse eigenvalue problem for SPD Toeplitz matrices, formulated as isospectral flows [4]. A different example is the tracing of nerve fibres in diffusion tensor imaging of the brain, where the voxels are symmetric positive definite diffusion tensors. ...
... As is well known, all RK methods preserve linear invariants [20], and symplectic RK methods preserve arbitrary quadratic invariants [16]. However, no RK methods can preserve arbitrary polynomial invariants of degree higher than two [11]. To utilize the quadratic invariant-preserving property of symplectic RK methods, we transform the logarithmic Hamiltonian functional to a quadratic functional by adopting the newly developed IEQ technique. ...
... In 1987, Cooper [8] showed that all RK methods conserve linear invariants and an irreducible RK method can preserve all quadratic invariants if and only if their coefficients satisfies the specific algebraic condition. Then, Calvo et al. [4] proved that no RK method can preserve arbitrary polynomial invariants of degree 3 or higher of arbitrary vector fields. Consequently, over the past few decades, there has been an increasing interest in higher-order energy-preserving methods for general conservative systems. ...
In this paper, we present a novel class of high-order energy-preserving schemes for solving the Zakharov-Rubenchik equations. The main idea of the scheme is first to introduce an quadratic auxiliary variable to transform the Hamiltonian energy into a modified quadratic energy and the original system is then reformulated into an equivalent system which satisfies the mass, modified energy as well as two linear invariants. The symplectic Runge-Kutta method in time, together with the Fourier pseudo-spectral method in space is employed to compute the solution of the reformulated system. The main benefit of the proposed schemes is that it can achieve arbitrarily high-order accurate in time and conserve the three invariants: mass, Hamiltonian energy and two linear invariants. In addition, an efficient fixed-point iteration is proposed to solve the resulting nonlinear equations of the proposed schemes. Several experiments are addressed to validate the theoretical results.
... , c s as the nodes of Gaussian quadrature formula, it yields Gauss-type RK (GRK) method or Gauss collocation method [20]. Since the RK methods are unable to guarantee all polynomial invariants of degree higher than two [21], it is therefore a great challenge to develop high-order energy-conserving schemes via the RK methods. ...
In this paper, we design two classes of high-accuracy conservative numerical algorithms for the nonlinear Klein-Gordon-Schrödinger system in two dimensions. By introducing the energy quadratization technique, we first transform the original system into an equivalent one, where the energy is modified as a quadratic form. The Gauss-type Runge-Kutta method and the Fourier pseudo-spectral method are then employed to discretize the reformulation system in time and space, respectively. The fully discrete schemes inherit the conservation of mass and modified energy and can reach high-order accuracy in both temporal and spatial directions. In order to complement the proposed schemes and speed up the calculation, we also develop another class of conservative schemes combined with the prediction-correction technique. Numerous experimental results are reported to demonstrate the efficiency and high accuracy of the new methods.
... In 1987, Cooper [8] showed that all RK methods conserve linear invariants and an irreducible RK method can preserve all quadratic invariants if and only if their coefficients satisfies the specific algebraic condition. Then, Calvo et al. [4] proved that no RK method can preserve arbitrary polynomial invariants of degree 3 or higher of arbitrary vector fields. Consequently, over the past few decades, there has been an increasing interest in higher-order energy-preserving methods for general conservative systems. ...
In this paper, we present a high-order energy-preserving scheme for solving Zakharov-Rubenchik equation. The main idea of the method is first to reformulate the original system into an equivalent one by introducing an quadratic auxiliary variable, and the symplectic Runge-Kutta method, together with the Fourier pseudo-spectral method is then employed to compute the solution of the reformulated system. The main benefit of the proposed method is that it can conserve the three invariants of the original system and achieves arbitrarily high-order accurate in time. In addition, an efficient fixed-point iteration is proposed to solve the resulting nonlinear equations of the proposed schemes. Several experiments are provided to validate the theoretical results.
... For more details, please refer to [29]. Second, it is known that any Runge-Kutta method conserves all linear invariants [6], however, we should note that the proposed linear momentum-preserving schemes cannot preserve the discrete mass since the time discretization is done using the extrapolation/prediction-correction technique. Although the higher order fully-implicit, mass and momentum conserving schemes can be obtained by applying a high-order symplectic Runge-Kutta method to the system (2.1) (see [15]), it is challenging to construct high-order linear ones which can preserve both the mass and momentum. ...
In this paper, a class of arbitrarily high-order linear momentum-preserving and energy-preserving schemes are proposed, respectively, for solving the regularized long-wave equation. For the momentum-preserving scheme, the key idea is based on the extrapolation/prediction-correction technique and the symplectic Runge-Kutta method in time, together with the standard Fourier pseudo-spectral method in space. We show that the scheme is linear, high-order, unconditionally stable and preserves the discrete momentum of the system. For the energy-preserving scheme, it is mainly based on the energy quadratization approach and the analogous linearized strategy used in the construction of the linear momentum-preserving scheme. The proposed scheme is linear, high-order and can preserve both the discrete mass and the discrete quadratic energy exactly. Numerical results are addressed to demonstrate the accuracy and efficiency of the proposed scheme.
... It has been proven that no Runge-Kutta (RK) method can preserve arbitrary polynomial invariants of degree 3 or higher of arbitrary vector fields [14]. Thus, over the last decades, how to develop high-order energy-preserving methods for general conservation systems attracts a lot of attention. ...
A novel class of high-order linearly implicit energy-preserving integrating factor Runge–Kutta methods are proposed for the nonlinear Schrödinger equation. Based on the idea of the scalar auxiliary variable approach, the original equation is first reformulated into an equivalent form which satisfies a quadratic energy. The spatial derivatives of the system are then approximated with the standard Fourier pseudo-spectral method. Subsequently, we apply the extrapolation technique/prediction–correction strategy to the nonlinear terms of the semi-discretized system and a linearized energy-conserving system is obtained. A fully discrete scheme is gained by further using the integrating factor Runge–Kutta method to the resulting system. We show that, under certain circumstances for the coefficients of a Runge–Kutta method, the proposed scheme can produce numerical solutions along which the modified energy is precisely conserved, as is the case with the analytical solution and is extremely efficient in the sense that only linear equations with constant coefficients need to be solved at every time step. Numerical results are addressed to demonstrate the remarkable superiority of the proposed schemes in comparison with other existing structure-preserving schemes.
In this paper we discuss numerical integration of the stationary Landau-Lifshitz (LL) equation. Using a Lax pair representation of the LL equation, we propose an isospectral algorithm that preserves the geometric structure of the system. The algorithm computes a discrete flow of a pair of matrices satisfying Lax-type equations and projects the flow on the phase space of the system. Since the stationary LL equation is equivalent to an integrable Hamiltonian system on the cotangent bundle of the unit sphere, we show that it can be also integrated by a symplectic method for constrained Hamiltonian systems. Comparison of the two methods demonstrates that they are similar in terms of accuracy and stability over long-time integration, but the isospectral method is much faster since it avoids solving a system of nonlinear equations required at each iteration of the symplectic algorithm.
The exponential fitting technique uses information on the expected behaviour of the solution of ordinary and partial differential equations to define accurate and efficient numerical methods. In particular, exponentially fitted methods are very effective when applied to problems with oscillatory solutions. In these cases, they perform better than standard methods, particularly in the large time-step regime.
In this paper we consider exponentially fitted Runge–Kutta methods and we give characterizations of those that preserve local conservation laws of linear and quadratic quantities.
As a benchmark for the general theory we apply the symplectic midpoint and its exponentially fitted version to approximate breather wave solutions of partial differential equations arising as models in several fields, such as fluid dynamics and quantum physics. Numerical tests show that the exponentially fitted method performs better and its solution exactly satisfy discrete conservation laws of mass (or charge) and momentum.
The problems of computing least squares approximations for various types of real and symmetric matrices subject to spectral constraints share a common structure. This paper describes a general procedure in using the projected gradient method. It is shown that the projected gradient of the objective function on the manifold of constraints usually can be formulated explicity. This gives rise to the construction of a descent flow that can be followed numerically. The explicit form also facilitates the computation of the second-order optimality conditions. Examples of applications are discussed. With slight modifications, the procedure can be extended to solve least squares problems for general matrices subject to singular-value constraints.
In this paper I shall consider the following problem: Given a symmetric (or Hermitian) matrix, find its eigenvalues. I shall use the theory of ordinary differential equations to solve this problem. In particular, I shall describe a spectrum-preserving dynamical system on symmetric (or Hermitian) matrices that flows “downhill” toward diagonal matrices. I shall show that diagonal matrices are the only stable equilibrium points of this flow.
In this paper we consider a variety of isospectral flows on the set of $n \times n$ matrices. These flows arise from Lax pairs and can all be interpreted in terms of the $QR$ decomposition for nonsingular matrices. The asymptotics of these differential equations are considered in detail and for symmetric matrices these asymptotics provide a new method of solving the eigenvalue problem.
The task of finding the singular-value decomposition (SVD) of a finite-dimen- sional complex Iinear operator is here addressed via gradient flows evolving on groups of complex unitary matrices and associated self-equivalent flows. The work constitutes a generalization of that of Brockett on the diagonalization of real symmetric matrices via gradient flows on orthogonal matrices and associated isospectral flows, It comple- ments results of Symes, Chu, and other authors on continuous analogs of the classical QR algorithm as well as earlier work by the authors on SVD via gradient flows on positive definite matrices.
A continuous version of the classical QR algorithm, known as the Toda flow, is generalized to complex-valued, full and nonsymmetric matrices. It is shown that this generalized Toda flow, when sampled at integer times, gives the same sequence of matrices as the OR algorithm applied to the matrix exp $(G(X_0 ))$. When $G(X) = X$, global convergence is deduced for the case of distinct real eigenvalues. This convergence property can also be understood locally by the center manifold theory. It is shown that the manifold of upper triangular matrices with decreasing main diagonal entries is the stable center manifold for the Toda flow. One interesting example is given to demonstrate geometrically the dynamical behavior of this flow.
In this paper the authors develop a general framework for calculating the eigenvalues of a symmetric matrix using ordinary differential equations. New algorithms are suggested and old algorithms, including $QR$, are interpreted.
The Toda flow is a dynamical system whose dependent variables may be viewed as the entries of a symmetric tridiagonal matrix. The spectrum of the matrix is invariant in time and, as $t \to \infty $, the off-diagonal entries tend to zero, exposing the eigenvalues on the main diagonal. Recently a wave of interest in the Toda and related isospectral flows was sparked in the numerical analysis community by the suggestion that to integrate the Toda flow numerically might be a cost effective way to calculate the eigenvalues of large symmetric tridiagonal matrices. A second source of interest was the recently discovered connection between the Toda flow and the $QR$ algorithm. The present paper has two principal aims. The first is to point out that the Toda and related flows are intimately connected with the power method. This connection clarifies completely the convergence properties of the flows. The relationship of the Toda flow to both the $QR$ algorithm and the power method is based on a connection with the $QR$ matrix factorization. The second aim of this paper is to introduce and discuss families of isospectral flows associated with two other well-known matrix factorizations, the $LU$ factorization and the Cholesky factorization.