Study Of Several Finite Difference Parallel Algorithm For Parabolic Equations

In many areas of natural sciences, such as heat conduction, other diffusion phenomena, certain biological morphology and chemical reactions, they are described through parabolic equation or equations. In some large complex scientific and engineering computing problems, they need fast computation. Reasonable parallel algorithm should be designed according to their internal parallelism, and then the answer will be obtained with parallel algorithm in parallel machine. Generally, these equations must be solved by the finite difference method. Therefore, those existed and traditional difference methods have to be constantly refined and perfected. New difference algorithm with parallelism should be constructed reasonably for specific questions.A simple one-dimensional parabolic equation is used as an example, Using Saul'yev Asymmetric format Among them, r =ΔΔxt2, GE grouped method is constructed, When r =ΔΔxt2≤1, GE is stable. Compared with the non-symmetric form alone, the truncation error has significantly improved, and the error is o (Δx+Δt).When GEL and GER are used alternatively at different time layers, AGE will be obtained. The description of Alternating Group Explicit is as follows:The research shows that AGE method is absolutely stable. The truncation errors are showed respectively in the following expressions:The truncation error of AGE is o (Δx+Δt).Then, the AGE method is extended to solve the two-dimensional finite difference parabolic equations in parallel computing, the method contains parallelism and is unconditionally stable. Take the initial boundary value problem of two-dimensional parabolic diffusion as an example: Boundary conditions are Initial condition is u ( x,y,0)= f(x,y) 0< x,y<1 AGE method can be defined asAlternating difference block methods is a new method used to solve two-dimensional problem inspired by the solution method of AGE difference. It makes use of the difference relationship between adjacent grid points to solve the problem. It includes the differential block Law and complementary difference block methods. Under certain conditions, this method is capable of doing parallel computing, and has good stability.For one-dimensional parabolic equation, according to the ideological group alternate format by Saul'yev asymmetric, a new difference algorithm with parallelism is established, which is generally described as: where Growth Matrix:Matrix G1 and G 2 set are non-negative. The new algorithm is absolutely stable according to Kellogg lemma.Finally, numerical experiments are done to compare the AGE algorithm and new algorithm, when r <1, =1 or > 1, the new algorithms are convergent and its errors are controllable. The previous theoretical analysis on the stability of the new algorithm is proved, namely, the new algorithm is absolutely stable.