This paper discusses a Python interface for the recently published Dune-Fem-DG module which provides highly efcient implementations of the discontinuous Galerkin(DG)method for solving a wide range of nonlinear partial...This paper discusses a Python interface for the recently published Dune-Fem-DG module which provides highly efcient implementations of the discontinuous Galerkin(DG)method for solving a wide range of nonlinear partial diferential equations(PDEs).Although the C++interfaces of Dune-Fem-DG are highly fexible and customizable,a solid knowledge of C++is necessary to make use of this powerful tool.With this work,easier user interfaces based on Python and the unifed form language are provided to open Dune-Fem-DG for a broader audience.The Python interfaces are demonstrated for both parabolic and frst-order hyperbolic PDEs.展开更多
In this paper, three implicit finite difference methods are developed to solve one dimensional time fractional advection-diffusion equation. The fractional derivative is treated by applying right shifted Grünwald...In this paper, three implicit finite difference methods are developed to solve one dimensional time fractional advection-diffusion equation. The fractional derivative is treated by applying right shifted Grünwald-Letnikov formula of order α ∈(0, 1). We investigate the stability analysis by using von Neumann method with mathematical induction and prove that these three proposed methods are unconditionally stable. Numerical results are presented to demonstrate the effectiveness of the schemes mentioned in this paper.展开更多
The main aim of this paper is to analyze the numerical method based upon the spectral element technique for the numerical solution of the fractional advection-diffusion equa-tion.The time variable has been discretized...The main aim of this paper is to analyze the numerical method based upon the spectral element technique for the numerical solution of the fractional advection-diffusion equa-tion.The time variable has been discretized by a second-order finite difference procedure.The stability and the convergence of the semi-discrete formula have been proven.Then,the spatial variable of the main PDEs is approximated by the spectral element method.The convergence order of the fully discrete scheme is studied.The basis functions of the spectral element method are based upon a class of Legendre polynomials.The numerical experiments confirm the theoretical results.展开更多
文摘This paper discusses a Python interface for the recently published Dune-Fem-DG module which provides highly efcient implementations of the discontinuous Galerkin(DG)method for solving a wide range of nonlinear partial diferential equations(PDEs).Although the C++interfaces of Dune-Fem-DG are highly fexible and customizable,a solid knowledge of C++is necessary to make use of this powerful tool.With this work,easier user interfaces based on Python and the unifed form language are provided to open Dune-Fem-DG for a broader audience.The Python interfaces are demonstrated for both parabolic and frst-order hyperbolic PDEs.
文摘In this paper, three implicit finite difference methods are developed to solve one dimensional time fractional advection-diffusion equation. The fractional derivative is treated by applying right shifted Grünwald-Letnikov formula of order α ∈(0, 1). We investigate the stability analysis by using von Neumann method with mathematical induction and prove that these three proposed methods are unconditionally stable. Numerical results are presented to demonstrate the effectiveness of the schemes mentioned in this paper.
基金The authors are grateful to the two reviewers for carefully reading this paper and for their comments and suggestions which have highly improved the paper.
文摘The main aim of this paper is to analyze the numerical method based upon the spectral element technique for the numerical solution of the fractional advection-diffusion equa-tion.The time variable has been discretized by a second-order finite difference procedure.The stability and the convergence of the semi-discrete formula have been proven.Then,the spatial variable of the main PDEs is approximated by the spectral element method.The convergence order of the fully discrete scheme is studied.The basis functions of the spectral element method are based upon a class of Legendre polynomials.The numerical experiments confirm the theoretical results.
