Parallel Finite Difference Method For Parabolic Equations

Parabolic partial differential equation,also called parabolic equation,is an im-portant partial differential equation.In many fields of science,many phenomena are described by parabolic equation(s),such as the diffusion of particles and energy,phys-ical chemistry reaction,biological migration and material interaction etc.Parabolic equations have been widely used in engineering.Numerous works have been devoted in developing efficient numerical methods for parabolic equations.Among the modern numerical methods,the finite difference method is the earliest method used by scientific and technological workers,and the theory of this method had developed.Therefore,the finite difference method has become an important method for solving parabolic equations.With the development of large-scale and parallel computers,traditional fi-nite difference methods have exposed many disadvantages for parabolic equation.For example,the time step of the classical explicit scheme is strictly limited.The classical implicit scheme can be solved only by linear equations,so it is not applicable to parallel computing.In this paper,some new parallel finite difference methods for parabolic equations are constructed,which has parallelism,stability and good calculation accu-racy.The main work of this paper can be divided into five chapters.Chapter 1 introduces the research background,research status as well as the article structure arrangement.In Chapter 2,the parallel finite difference method for heat equation is constructed.The new parallel algorithm consists of two DDMs.Each one is applied to compute the values at(n+1)st time level by use of known numerical solutions at n-th time level,respectively.Then the average of two above values is chosen to be the numerical solutions at(n +1)st time level.Compared with classical finite difference method,the new algorithm obtains satisfactory accuracy while maintaining parallelism and unconditional stability.This algorithm can be extended to solve two-dimensional heat equations by alternating direction implicit(ADI)technique.Both theory analysis and numerical experiment show the efficiency of the new algorithm.In Chapter 3,two parallel algorithms that based on modified Crank-Nicolson scheme and Samarskii scheme for convection-dominated diffusion equation are proposed,respectively.Two algorithms are better than the previous finite difference method in reducing the numerical oscillations of convection-dominated diffusion equation.The second-order convergence speed and unconditional stability of two algorithms.Numer-ical experiments show the efficiencies of two new algorithms.In Chapter 4,an alternating segment explicit-implicit scheme for Burgers equa-tion is constructed.The new parallel algorithm makes the explicit scheme and implicit scheme combined by Saul'yev asymmetric schemes.The new algorithm is three level schemes that are applied to solve the numerical solutions by alternating technique at this time level and next time level.The space segments can be divided into multiple sub-domains for parallel computing.The new algorithm has parallelism and it is un-conditionally stable.Numerical experiment is given to show the efficiency of the new algorithm.The last chapter gives the summary and prospects for future work.