A High Order Parallel Difference Scheme Of Parabolic Equation

In this paper, a high order parallel finite difference algorithm of parabolic equation is presented. Firstly, we combine the values of previous three levels at the interface points with the compact scheme to solve the values of sub-domains in parallel, then the values at the interface points are computed by the compact scheme. For one-dimensional scheme. the stability bound of the procedure is derived to be at least (?), Fur two-dimensional scheme, it is (?) , and the convergence rate is proved to be of order four. Numerical examples show that this method has much better accuracy than other known methods.The paper is composed of three chapters as follows:In the first chapter, we mainly introduce the background theory of parallel finite difference scheme for parabolic equation, and some results of the method we use.The second chapter is composed of three parts:In the first part,we give the model of one-dimensional heat equation discussed in this paper:then we divide the domain [0,l]×[0. T] into small grids with the time step lengthτ and the space step length h,and wo also give a parallel finite difference scheme based on the compact scheme.In the second part,by the discrete poincar(?) inequality and the discrete Green formula,the stability bound of the procedure is derived to be at least (?),and the convergence rate of the numerical solution is proved to be of order four.In the third part,we give the numerical examples to proof the stability of the difference scheme and the stability bound of the procedure,we also compare the accuracy of the difference scheme presented in this paper with that of the classical explicit scheme,thc AGE method, the domain decomposition algorithm presented in [7],it shows that the method in this paper has much better accuracy.The third chapter is also composed of three parts:In the first part,we give the model of two-dimensional heat equation discussed in this paper:andΩ={(x,y)|0＜x＜1,0＜y＜1},we divide the domainΩ×[0,T] into small grids with the time step lengthτand the space step length h,then gives the corresponding parallel finite difference scheme.In the second part,we prove the convergence rate of the two-dimensional problem,analyze the stability of the scheme.The stability bound of the procedure is derivedto be at least(?).In the third part,numerical examples are proposed to proof the stability and the accuracy of the difference scheme.