In this paper,a class of discrete Gronwall inequalities is proposed.It is efciently applied to analyzing the constructed L1/local discontinuous Galerkin(LDG)fnite element meth-ods which are used for numerically solvin...In this paper,a class of discrete Gronwall inequalities is proposed.It is efciently applied to analyzing the constructed L1/local discontinuous Galerkin(LDG)fnite element meth-ods which are used for numerically solving the Caputo-Hadamard time fractional difusion equation.The derived numerical methods are shown to beα-robust using the newly estab-lished Gronwall inequalities,that is,it remains valid whenα→1^(−).Numerical experiments are given to demonstrate the theoretical statements.展开更多
For two-dimensional(2D)time fractional diffusion equations,we construct a numerical method based on a local discontinuous Galerkin(LDG)method in space and a finite differ-ence scheme in time.We investigate the numeric...For two-dimensional(2D)time fractional diffusion equations,we construct a numerical method based on a local discontinuous Galerkin(LDG)method in space and a finite differ-ence scheme in time.We investigate the numerical stability and convergence of the method for both rectangular and triangular meshes and show that the method is unconditionally stable.Numerical results indicate the effectiveness and accuracy of the method and con-firm the analysis.展开更多
In this paper,an efficient method is proposed to solve the Caputo diffusion equation with a variable coefficient.Since the solution of such an equation in general has a typical weak singularity near the initial time t...In this paper,an efficient method is proposed to solve the Caputo diffusion equation with a variable coefficient.Since the solution of such an equation in general has a typical weak singularity near the initial time t=0,the time-fractional derivative with order in(0,1)is discretized by L2-1_(σ)formula on nonuniform meshes.For the spatial derivative,the local discontinuous Galerkin(LDG)method is employed.A complete theoretical analysis of the numerical stability and convergence of the derived scheme is given using a discrete fractional Gronwall inequality.Numerical experiments demonstrate the validity of the established scheme and the accuracy of the theoretical analysis results.展开更多
For the high-order diffusion and dispersion equations, the general practice of the explicit-implicit-null (EIN) method is to add and subtract an appropriately large linear highest derivative term with a constant coeff...For the high-order diffusion and dispersion equations, the general practice of the explicit-implicit-null (EIN) method is to add and subtract an appropriately large linear highest derivative term with a constant coefficient at one side of the equation, and then apply the standard implicit-explicit method to the equivalent equation. We call this approach the constant-coefficient EIN method in this paper and hereafter denote it by “CC-EIN”. To reduce the error in the CC-EIN method, the variable-coefficient explicit-implicit-null (VC-EIN) method, which is obtained by adding and subtracting a linear highest derivative term with a variable coefficient, is proposed and studied in this paper. Coupled with the local discontinuous Galerkin (LDG) spatial discretization, the VC-EIN method is shown to be unconditionally stable and can achieve high order of accuracy for both one-dimensional and two-dimensional quasi-linear and nonlinear equations. In addition, although the computational cost slightly increases, the VC-EIN method can obtain more accurate results than the CC-EIN method, if the diffusion coefficient or the dispersion coefficient has a few high and narrow bumps and the bumps only account for a small part of the whole computational domain.展开更多
In this article,some high-order local discontinuous Galerkin(LDG)schemes based on some second-order θ approximation formulas in time are presented to solve a two-dimen-sional nonlinear fractional diffusion equation.T...In this article,some high-order local discontinuous Galerkin(LDG)schemes based on some second-order θ approximation formulas in time are presented to solve a two-dimen-sional nonlinear fractional diffusion equation.The unconditional stability of the LDG scheme is proved,and an a priori error estimate with O(h^(k+1)+At^(2))is derived,where k≥0 denotes the index of the basis function.Extensive numerical results with Q^(k)(k=0,1,2,3)elements are provided to confirm our theoretical results,which also show that the second-order convergence rate in time is not impacted by the changed parameter θ.展开更多
This paper is concerned with numerical solutions of the LDG method for 1D wave equations.Superconvergence and energy conserving properties have been studied.We first study the superconvergence phenomenon for linear pr...This paper is concerned with numerical solutions of the LDG method for 1D wave equations.Superconvergence and energy conserving properties have been studied.We first study the superconvergence phenomenon for linear problems when alternating fluxes are used.We prove that,under some proper initial discretization,the numerical trace of the LDG approximation at nodes,as well as the cell average,converge with an order 2k+1.In addition,we establish k+2-th order and k+1-th order superconvergence rates for the function value error and the derivative error at Radau points,respectively.As a byproduct,we prove that the LDG solution is superconvergent with an order k+2 towards the Radau projection of the exact solution.Numerical experiments demonstrate that in most cases,our error estimates are optimal,i.e.,the error bounds are sharp.In the second part,we propose a fully discrete numerical scheme that conserves the discrete energy.Due to the energy conserving property,after long time integration,our method still stays accurate when applied to nonlinear Klein-Gordon and Sine-Gordon equations.展开更多
基金supported by the National Natural Science Foundation of China under Grant No.12101266.
文摘In this paper,a class of discrete Gronwall inequalities is proposed.It is efciently applied to analyzing the constructed L1/local discontinuous Galerkin(LDG)fnite element meth-ods which are used for numerically solving the Caputo-Hadamard time fractional difusion equation.The derived numerical methods are shown to beα-robust using the newly estab-lished Gronwall inequalities,that is,it remains valid whenα→1^(−).Numerical experiments are given to demonstrate the theoretical statements.
文摘For two-dimensional(2D)time fractional diffusion equations,we construct a numerical method based on a local discontinuous Galerkin(LDG)method in space and a finite differ-ence scheme in time.We investigate the numerical stability and convergence of the method for both rectangular and triangular meshes and show that the method is unconditionally stable.Numerical results indicate the effectiveness and accuracy of the method and con-firm the analysis.
文摘In this paper,an efficient method is proposed to solve the Caputo diffusion equation with a variable coefficient.Since the solution of such an equation in general has a typical weak singularity near the initial time t=0,the time-fractional derivative with order in(0,1)is discretized by L2-1_(σ)formula on nonuniform meshes.For the spatial derivative,the local discontinuous Galerkin(LDG)method is employed.A complete theoretical analysis of the numerical stability and convergence of the derived scheme is given using a discrete fractional Gronwall inequality.Numerical experiments demonstrate the validity of the established scheme and the accuracy of the theoretical analysis results.
基金supported in part by the National Key R&D Program of China No.2023YFA1009003 and the NSFC grant 12031001Chi-Wang Shu research is supported in part by the NSF grants DMS-2010107 and DMS-2309249.
文摘For the high-order diffusion and dispersion equations, the general practice of the explicit-implicit-null (EIN) method is to add and subtract an appropriately large linear highest derivative term with a constant coefficient at one side of the equation, and then apply the standard implicit-explicit method to the equivalent equation. We call this approach the constant-coefficient EIN method in this paper and hereafter denote it by “CC-EIN”. To reduce the error in the CC-EIN method, the variable-coefficient explicit-implicit-null (VC-EIN) method, which is obtained by adding and subtracting a linear highest derivative term with a variable coefficient, is proposed and studied in this paper. Coupled with the local discontinuous Galerkin (LDG) spatial discretization, the VC-EIN method is shown to be unconditionally stable and can achieve high order of accuracy for both one-dimensional and two-dimensional quasi-linear and nonlinear equations. In addition, although the computational cost slightly increases, the VC-EIN method can obtain more accurate results than the CC-EIN method, if the diffusion coefficient or the dispersion coefficient has a few high and narrow bumps and the bumps only account for a small part of the whole computational domain.
基金This work is supported by the National Natural Science Foundation of China(11661058,11761053)the Natural Science Foundation of Inner Mongolia(2017MS0107)the Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region(NJYT-17-A07).
文摘In this article,some high-order local discontinuous Galerkin(LDG)schemes based on some second-order θ approximation formulas in time are presented to solve a two-dimen-sional nonlinear fractional diffusion equation.The unconditional stability of the LDG scheme is proved,and an a priori error estimate with O(h^(k+1)+At^(2))is derived,where k≥0 denotes the index of the basis function.Extensive numerical results with Q^(k)(k=0,1,2,3)elements are provided to confirm our theoretical results,which also show that the second-order convergence rate in time is not impacted by the changed parameter θ.
基金supported by Natural Science Foundation of Jiangsu Province(No.BK20170374)Nature Science Research Program for Colleges and Universities of Jiangsu Province(No.17KJB110016)Scientific Research Project for University Students of Suzhou University of Science and Technology in 2017-2018
基金This work is supported in part by the National Natural Science Foundation of China(NSFC)under grants Nos.11201161,11471031,11501026,91430216,U1530401China Postdoctoral Science Foundation under grant Nos.2015M570026,2016T90027the US National Science Foundation(NSF)through grant DMS-1419040。
文摘This paper is concerned with numerical solutions of the LDG method for 1D wave equations.Superconvergence and energy conserving properties have been studied.We first study the superconvergence phenomenon for linear problems when alternating fluxes are used.We prove that,under some proper initial discretization,the numerical trace of the LDG approximation at nodes,as well as the cell average,converge with an order 2k+1.In addition,we establish k+2-th order and k+1-th order superconvergence rates for the function value error and the derivative error at Radau points,respectively.As a byproduct,we prove that the LDG solution is superconvergent with an order k+2 towards the Radau projection of the exact solution.Numerical experiments demonstrate that in most cases,our error estimates are optimal,i.e.,the error bounds are sharp.In the second part,we propose a fully discrete numerical scheme that conserves the discrete energy.Due to the energy conserving property,after long time integration,our method still stays accurate when applied to nonlinear Klein-Gordon and Sine-Gordon equations.
