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RBF-based meshless local Petrov Galerkin method for the multi-dimensional convection–diffusion-reaction equation
... Numerical solution of CDEs has become an important means of analysis. Robust numerical methods for solving CDEs are the Boundary Element Method (ΒΕΜ) [3], [4], the Finite Volume Method (FVM) [5], [6], the Finite Difference Method (FDM) [7], [8], the Finite Element Method (FEM) [9]- [11], and the Mesh Free Method (MFM) [12], [13]. Although these methods can be used to solve a wide range of engineering problems in solid mechanics, fluid mechanics and heat transfer, they also have some drawbacks. ...
... In eqn (12), the first term is the convection term, and the second one is the diffusion term. When we discretize eqn (12) in FLM, we need to employ two different line sets for these two terms, the upwind line set and the central line set (the central line set also being called the diffusion line set), respectively. ...
... In eqn (12), the first term is the convection term, and the second one is the diffusion term. When we discretize eqn (12) in FLM, we need to employ two different line sets for these two terms, the upwind line set and the central line set (the central line set also being called the diffusion line set), respectively. The former embodies the influence of the incoming flow on the collocation point, while the latter embodies the influence of all surrounding nodes on the collocation point. ...
... It has wide applications in ecological environment, fluid mechanics, biological mathematics, and other fields of natural science. For example, the CDR equation has been used to describe the following: the conduction of heat in the fluid [1], thermo-hygro transfer in porous media [2], predatorprey interactions in population densities [3], the transport of adsorbing contaminants and microbe-nutrient systems in groundwater [4], heat transfer in a draining film [5], etc. However, in most cases, similar to some other widely used mathematical models [6,7], the CDR equation cannot obtain exact solutions. ...
... For example, a local discrete exterior calculus discretization [8] of the convection diffusion equation for compressible and incompressible flow is proposed, and the discretization needs to be stabilized by introducing artificial diffusion. For the CDR equation, the numerical methods mainly include finite element method [9][10][11][12][13][14], integration factor method [15][16][17][18], meshless method [1,19], finite difference (FD) method [20][21][22][23][24][25][26][27][28], and so on. Among them, the FD method is a traditional numerical method, which has been widely used in solving various fluid dynamic equations for a long time [29]. ...
... Firstly, we consider the 2D unsteady CDR equation [14,18] with variable coefficients as follows: u t + p(x, y, t)u x +q(x, y, t)u y +c(x, y, t)u = α(u xx +u yy )+ f (x, y, t), (x, y) ∈ Ω, t ≥ 0, (1) with initial and boundary conditions u(x, y, 0) = g(x, y), (x, y) ∈ Ω, (2) u(x, y, t) = s(x, y, t), (x, y) ∈ ∂Ω, t > 0. ...
In this paper, a type of high-order compact (HOC) finite difference method is developed for solving two- and three-dimensional unsteady convection diffusion reaction (CDR) equations with variable coefficients. Firstly, an HOC difference scheme is derived to solve the two-dimensional (2D) unsteady CDR equation. Discretization in time is carried out by Taylor series expansion and correction of the truncation error remainder, while discretization in space is based on the fourth-order compact difference formulas. The scheme is second-order accuracy in time and fourth-order accuracy in space. The unconditional stability is obtained by the von Neumann analysis method. Then, this scheme is extended to solve the three-dimensional (3D) unsteady CDR equation. It needs only a five-point stencil for 2D problems and a seven-point stencil for 3D problems. Moreover, the present schemes can solve the nonlinear Burgers equation. Finally, numerical experiments are conducted to show the good performances of the new schemes.
... How to effectively solve nonlinear partial differential equations has been a long-standing concern for researchers, and various numerical methods for solving this equation include finite volume method, finite element method, lattice Boltzmann method, spectral method, finite difference method, etc. [10][11][12][13][14][15][16][17][18][19]. The finite difference method is widely used due to its simplicity, strong operability, easy access to high order numerical formats, and ease of programming. ...
In this paper, we investigate the numerical computation method for a one-dimensional self diffusion plant water model with homogeneous Neumann boundary conditions. First, a high accuracy compact difference scheme for the diffusive plant water model in an arid flat environment is constructed using the finite difference method. The fourth order compact difference scheme is used for the spatial derivative term, and the Taylor series expansion and residual correction function are used to discretize the time term. We obtain a difference scheme with second-order accuracy in time and fourth-order accuracy in space. Second, the Fourier analysis method is used to prove that the above format is unconditionally stable. Then, the numerical examples provided the convergence and accuracy of the difference scheme. Finally, numerical simulations are conducted near the Turing Hopf bifurcation point of the model to obtain the spatial distribution maps of vegetation and water under small disturbances of different parameters. In this paper, the evolution law of vegetation quantity and water density at any time is observed.Revealing the impact of small changes in parameters on the spatiotemporal dynamics of plant water models will provide a basis for understanding whether ecosystems are fragile.
... In recent years, several meshless methods have been used to analyze convectiondiffusion-reaction problems, such as the Galerkin and least squares method [11], variational multiscale element-free Galerkin methods [12][13][14], meshfree local Petrov Galerkin methods [15,16], Hermite method [17], and local knot method [18]. ...
In order to obtain the numerical results of 3D convection-diffusion-reaction problems with variable coefficients efficiently, we select the improved element-free Galerkin (IEFG) method instead of the traditional element-free Galerkin (EFG) method by using the improved moving least-squares (MLS) approximation to obtain the shape function. For the governing equation of 3D convection-diffusion-reaction problems, we can derive the corresponding equivalent functional; then, the essential boundary conditions are imposed by applying the penalty method; thus, the equivalent integral weak form is obtained. By introducing the IMLS approximation, we can derive the final solved linear equations of the convection-diffusion-reaction problem. In numerical examples, the scale parameter and the penalty factor of the IEFG method for such problems are discussed, the convergence is proved numerically, and the calculation efficiency of the IEFG method are verified by four numerical examples.
... These oscillations can be avoided through adaptive mesh refinement or by introducing stabilization techniques. The recently proposed Virtual Element Method [5,68] as well as meshless methods such as the MLPG [34] and the MFS [37] provide additional flexibility in the discretization of complex geometries and can efficiently capture the boundary and interior layers appearing in the numerical solution of transient CDR problems. ...
The convection-diffusion-reaction (CDR) equation has been extensively used to simulate a variety of physical phenomena. A robust numerical method for solving linear CDR problems is the Boundary Element Method (ΒΕΜ). However, the conventional BEM leads to dense coefficient matrices and as a result the memory requirements grow quadratically with respect to the number of degrees of freedom. In this work, a Local Domain BEM (LD-BEM) for solving the transient CDR equation with a constant velocity field is presented. The domain of interest is fragmented into small subdomains and the integral representation of the solution is considered separately for each of the subdomains. Eliminating the fluxes at all subdomain interfaces, the proposed LD-BEM leads to sparse linear system coefficient matrices and a reduced number of degrees of freedom. Eight numerical examples are solved to assess the efficiency and accuracy of the proposed method.
... Indeed, there are few theoretical limits to their ability to interpolate over arbitrary sets of points [10]. Owing to their flexibility, several schemes -such as a mesh-free Petrov-Galerkin method which made use of the MQ-RBF as a basis [5,13,34] -have been developed to exploit their properties. Other works have studied the properties of RBFs in the approximation of functions in various spaces and with reference to Discontinuous Galerkin, such as Wendland [56] and Wendland [57], and these have shown that RBFs can be used to form an effective approximation space under certain conditions. ...
Flux reconstruction provides a framework for solving partial differential equations in which functions are discontinuously approximated within elements. Typically, this is done by using polynomials. Here, the use of radial basis functions as a methods for underlying functional approximation is explored in one dimension, using both analytical and numerical methods. At some mesh densities, RBF flux reconstruction is found to outperform polynomial flux reconstruction, and this range of mesh densities becomes finer as the width of the RBF interpolator is increased. A method which avoids the poor conditioning of flat RBFs is used to test a wide range of basis shapes, and at very small values, the polynomial behaviour is recovered. Changing the location of the solution points is found to have an effect similar to that in polynomial FR, with the Gauss--Legendre points being the most effective. Altering the location of the functional centres is found to have only a very small effect on performance. Similar behaviours are determined for the non-linear Burgers' equation.
In this paper, we prove the boundedness of generalized fractional maximal operator \(M_{\rho }\) and generalized Riesz potential operator \(I_{\rho }\) in the modified Morrey spaces \(\widetilde{L}_{p,\lambda }({\mathbb {R}^n})\). We show that the sufficient conditions for the boundedness of the operator \(M_{\rho }\) and the operator \(I_{\rho }\) from the modified Morrey spaces \(\widetilde{L}_{p,\lambda }({\mathbb {R}^n})\) to another one \(\widetilde{L}_{q,\lambda }({\mathbb {R}^n})\), for \(1< p<q<\infty \) and from \(\widetilde{L}_{1,\lambda }({\mathbb {R}^n})\) to the weak modified Morrey spaces \(W\widetilde{L}_{q,\lambda }({\mathbb {R}^n})\), for \(p=1, 1<q<\infty \). We get the boundedness of our two-operators \(I_{\rho }\) and \(M_{\rho }\) in the modified Morrey spaces \(\widetilde{L}_{p,\lambda }({\mathbb {R}^n})\) using the local estimate given in the Lemma 2.KeywordsGeneralized Riesz potential operatorGeneralized fractional maximal operatorModified Morrey spaces
In this study, two-dimensional, unsteady convection-diffusion and convection-diffusion-reaction equations are numerically investigated with different time schemes. In the governing equations, the space derivatives are approximated by differential quadrature method (DQM) which gives highly accurate results using small number of grid points and the time derivatives are handled by different time schemes. The distribution of nodes is achieved by non-uniform Gauss-Chebyshev-Lobatto (GCL) grid points. The problems having the exact solutions are chosen to check the best error behavior. Computational cost in view of central processing unit time and the efficiency of different time schemes in terms of errors are examined. As expected, explicit time schemes need smaller time increments while implicit time schemes enable one to use larger time increments. In each chosen problems, Adams-Bashforth-Moulton and Runge-Kutta of order four exhibit the best error behavior.KeywordsConvection-diffusion-reactionDifferential quadrature methodTime schemes
By introducing the dimension splitting method (DSM) into the generalized element-free Galerkin (GEFG) method, a dimension splitting generalized interpolating element-free Galerkin (DS-GIEFG) method is presented for analyzing the numerical solutions of the singularly perturbed steady convection–diffusion–reaction (CDR) problems. In the DS-GIEFG method, the DSM is used to divide the two-dimensional CDR problem into a series of lower-dimensional problems. The GEFG and the improved interpolated moving least squares (IIMLS) methods are used to obtain the discrete equations on the subdivision plane. Finally, the IIMLS method is applied to assemble the discrete equations of the entire problem. Some examples are solved to verify the effectiveness of the DS-GIEFG method. The numerical results show that the numerical solution converges to the analytical solution with the decrease in node spacing, and the DS-GIEFG method has high computational efficiency and accuracy.
Radial basis function-generated finite difference method (RBF-FD) has been a popular method for simulating the derivatives of a function and has been successfully applied for the partial differential equations (PDEs). In this paper we introduce an effective h-adaptive RBF-FD method to the convection-diffusion equation in high-dimension space including two dimensions and three dimensions. The derivative of the solution is represented on overlapping a new influence domain through RBF-FD by using Thin Plane Spline Radial Basis Functions (TPS) augmented with additional polynomial functions. The number of the nodes added in the domain by the h-adaptive RBF-FD method is triggered by an error indicator, which very simply depends on the local residual norm. Several numerical examples are given to demonstrate the validity of h-adaptive RBF-FD method for the convection-diffusion equation in high-dimension space.
Spurious oscillations occur in the numerical results of convective dominated problems with discontinuities when central interpo- lation schemes are used. Upwind interpolation is necessary to overcome these oscillations. In this work, an upwind scheme is proposed to solve the unsteady convection-diffusion equation using Radial Basis Function (RBF) based Local Hermitian Interpo- lation (LHI) with PDE centres. It is a meshless numerical method based on the interpolation of overlapping RBF systems and also satisfy the governing partial differential equations at certain specified locations. Local hermite interpolation gives symmetric non-singular interpolation matrix which can be inverted easily to solve the system of equations. The scheme is implemented to unsteady benchmark problems and validated against the corresponding analytical solutions. Comparisons with the benchmark solutions show that the proposed upwind scheme is stable and produces significantly accurate results especially for convection dominated problems.
In this paper, a radial basis function finite difference (RBF-FD) method is proposed to solve the time fractional convection-diffusion equation in high dimensional space. Compact support Wendland RBF augmented with the polynomial terms is employed to determine a RBF-FD weights on neighboring nodes surrounding the center point for the spatial derivatives. And the time discretization is replaced by the shifted Grünwald formula. The method can lead to second-order convergence in space and time simultaneously. We get an efficient shape parameter by using an adaptive strategy to select the domain of the support. The main idea is to take into account the way the data points are distributed within the support domain in which the number of points is about constant. Moreover, the unconditional stability and convergence of the proposed method are deduced. Finally, numerical examples present that the proposed method leads to accurate results, fast evaluation, high efficiency for modeling and simulating for the fractional differential equations.
We investigate a novel method for the numerical solution of two-dimensional time-dependent convection-diffusion-reaction equations with nonhomogeneous boundary conditions. We first approximate the equation in space by a stable Gaussian radial basis function (RBF) method and obtain a matrix system of ODEs. The advantage of our method is that, by avoiding Kronecker products, this system can be solved using one of the standard methods for ODEs. For the linear case, we show that the matrix system of ODEs becomes a Sylvester-type equation, and for the nonlinear case we solve it using predictor-corrector schemes such as Adams-Bashforth and implicit-explicit (IMEX) methods. This work is based on the idea proposed in our previous paper (2016), in which we enhanced the expansion approach based on Hermite polynomials for evaluating Gaussian radial basis function interpolants. In the present paper the eigenfunction expansions are rebuilt based on Chebyshev polynomials which are more suitable in numerical computations. The accuracy, robustness and computational efficiency of the method are presented by numerically solving several problems.
In this article, we propose a new two-level method for solving 2D and 3D semilinear elliptic boundary value problem base on radial basis function (RBF). In the first step, we use the global RBF method to solve a semilinear problem on the coarse mesh or a small number of collocation points, in the second step, finite difference (FD), finite element (FE), and radial basis function-generated finite difference (RBF-FD) methods are used to solve the linearized problem on the fine mesh or a large number of collocation points, respectively. Numerical examples are provided to verify the feasibility and efficiency of the two-level RBF method. Moreover, compared to the FD and FE methods, RBF-FD method used in the second step further improve the accuracy of numerical solution.
In this paper, the localized radial basis function collocation method (LRBFCM) is combined with the partial upwind scheme for solving convection-dominated fluid flow problems. The localization technique adopted in LRBFCM has shown to be effective in avoiding the well known ill-conditioning problem of traditional meshless collocation method with globally defined radial basis functions (RBFs). For convection–diffusion problems with dominated convection, stiffness in the form of boundary/interior layers and shock waves emerge as convection overwhelms diffusion. We show in this paper that these kinds of stiff problems can be well tackled by combining the LRBFCM with partial upwind scheme. For verification, several numerical examples are given to demonstrate that this scheme improves the LRBFCM in providing stable, accurate, and oscillation-free solutions to one- and two-dimensional Burgers׳ equations with shock waves and singular perturbation problems with turning points and boundary layers.
