The Local Discontinuous Galerkin Method For Partial Differential Equation And Parallel Implementation

Local Discontinuous Galerkin method is the development of the Runge-Kutta discontinuous Galerkin method,due to its good charactcristics.it was well developed in recent years.This paper is divided into two sections,in section1of this paper we ap-ply the LDG method to solve fractional differential equations.in section2of this paper we describes a parallel implementation of the local discontinuous Galerkin method.In this paper, we apply the LDG method to solve a fractional differential equations:-Dx3u(x)=-(?)3u(x)/(?)xH=∫(x) in Ω=(a,b) u(a)=0, u(b)=c Here/(x) is source term,β∈[1-2].Riemann-Liouville definition of fractional derivative is first introduced.and then we give the weak form of fractional differential equations,and finally nu-merical experiments are presented.The LDG method is extremely local in data communications, thus allow-ing for efficient parallel implementations.We study the following two differential equations to analysis the LDG methods of parallel implementation.One-dimensional conservation equations:(?)u/(?)t+(?)u/(?)x=0,(x,t)∈[0,2]×[0,T] with initial condition u(x,0)—sin πx and periodic boundary conditions.Heat equation:(?)u/(?)t=(?)2u/(?)x2,(x,t)∈[0,2]×[0,T] with initial condition u(x,0)=sin πx and periodic boundary conditions.We first give a theoretical analysis, and then a numerical model, and finally numerical experiments are presented.The last section is the conclusion of this paper.