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 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
In this paper,L1/local discontinuous Galerkin method seeking the numerical solution to the time-fractional Stokes equation is displayed,where the timefractional derivative is in the sense of Caputo with derivative or...In this paper,L1/local discontinuous Galerkin method seeking the numerical solution to the time-fractional Stokes equation is displayed,where the timefractional derivative is in the sense of Caputo with derivative orderα∈(0,1).Although the time-fractional derivative is used,its solution may be smooth since such examples can be easily constructed.In this case,we use the uniform L1 scheme to approach the temporal derivative and use the local discontinuous Galerkin(LDG)method to approximate the spatial derivative.If the solution has a certain weak regularity at the initial time,we use the non-uniform L1 scheme to discretize the time derivative and still apply LDG method to discretizing the spatial derivative.The numerical stability and error analysis for both situations are studied.Numerical experiments are also presented which support the theoretical analysis.展开更多
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.
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.
基金supported by the National Natural Science Foundation of China(Grant Nos.12271339,12101266).
文摘In this paper,L1/local discontinuous Galerkin method seeking the numerical solution to the time-fractional Stokes equation is displayed,where the timefractional derivative is in the sense of Caputo with derivative orderα∈(0,1).Although the time-fractional derivative is used,its solution may be smooth since such examples can be easily constructed.In this case,we use the uniform L1 scheme to approach the temporal derivative and use the local discontinuous Galerkin(LDG)method to approximate the spatial derivative.If the solution has a certain weak regularity at the initial time,we use the non-uniform L1 scheme to discretize the time derivative and still apply LDG method to discretizing the spatial derivative.The numerical stability and error analysis for both situations are studied.Numerical experiments are also presented which support the theoretical analysis.
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
