Theory Of The Finite Difference Parallel Algorithms For Parabolic Equations

In science field, many phenomena are described by parabolic equation or parabolic system. Hence, numerically solving parabolic partial differential equation by finite difference method is significant in theory and application.With the appearance and the increasing availability of vector computers and parallel computers, traditional finite difference methods are being tested by large scale computation, parallelism and accuracy of all kinds of difference schemes must be compared in the parallel environment. Up to now, most of work consider the regular domain on which they construct parallel difference schemes generally by use of rectangular mesh. As to construction of parallel finite difference schemes and theoretical analysis on triangular mesh, research has little to be done. But in many practical problems, it is always convenient to take triangular mesh. In this paper, we will pay much attention to constructing practical and effective parallel finite difference methods on triangular mesh, in the meantime, we develop parallel finite difference methods both in one-dimensional case and in two-dimensional case on rectangular mesh.Motivated by the existing results, we mainly adopt two techniques, i.e., three-level alternating and domain decomposing to achieve parallel finite difference schemes for solving parabolic equation on triangular mesh. Due to the geometry complexity of triangular rnesh itself, more mesh points will be1Abstract 2involved in the finite difference equation. It brings about great difficulties in the implementation of the parallel computing for the finite difference schemes. In this paper, the main results are as follows.1. Three-level Alternating MethodsConsider the initial-boundary value problem of two-dimensional diffusion equation as followswhere is a rhombus. is an acute angle between the two neighboring sides of the rhombus. We superimpose the structured triangular mesh on , the mesh points are denoted by pj. Let h be a positive parameter related to the step size in space and t be the step size in time. Denote the mesh points on the n-th time level by (pij, tn),tn = nt, and the approximation of the exact solution for Eq.(l) at the points (p, tn) by . In the beginning, we construct some basic finite difference schemes on triangular mesh, such as explicit scheme, implicit scheme, Crank-Nicolson scheme, asymmetric scheme a) and asymmetric scheme b).(1) ABd(Alternating Band) MethodSuppose the number N - 1 of the interior points of domain in the .-direction is odd. If we use the asymmetric scheme a) and b) successively at interior points one after another along x- direction, we have to use asymmetric scheme a) at interior points adjacent to the right boundary alone. Then, we obtain the BdR(Band Right) method. In the procedure mentioned above, if we change asymmetric scheme a) into b), and change b) into a), we obtain the BdL(Banl Left) method.Abstract 3On the stabilities of the BdR and BdL method, we have the following results.Theorem 1 For , BdR and BdL method are stable.If we use the BdR and BdL method alternately on the odd level and the even level in time, we get the ABd method. On the stability of the ABd method, we have the following result.Theorem 2 ABd method is absolutely stable.(2) ABdE-I(Alternating Band Explicit-Implicit) MethodIn the process of constructing the ABd method, if we use implicit schemes at interior points between the points at which we use the asymmetric scheme a) and the asymmetric scheme b), the computation on a band region will be completely separate. We call the scheme on the band region as BIS (Band Implicit Segment). Utilizing the BIS, ABdE-I method is described as follows: for a given odd integer S, suppose there exists L 3 such that it satisfies N ?I = SL. According to band, we divide the points on domain into S segment...