Parallel Algorithms And Programs For Solving Parabolic Equations

This article mainly gives the parallel programs for solving one-dimensional and two-dimensional parabolic equations.The article first introduces the software environment.hardware environment and the latest developments of the parallel computers.Then it summarizes some difference shemes for the parabolic equations.including AGE method.ASE method for one-dimensional problem and AGE method for two-dimensional problem.Finally,it gives the speed-up ratio and efficiencyof the Algorithms through the results of the experiments.The article could be divided into three parts:brief introduction to Parallel computers,introduction to algorithms for solving parabolic equations,parallel programs of the algorithms and the analysis of the results among which the last part is the main work of the author. First consider the one-dimensional problem:We are able to get the GE scheme from (3.10).that is:Through alternately using the GER method and GEL method in different time levels,we get AGE method:G₁,G₂are defined in section(2.1).AGE method is absolutely stable.Then consider the two-dimensional diffusion problem: the initial condition is:the boundary value condition is:The GE scheme of (3.13)is:Through alternately using the GER method and GEL method in different time levels,we get AGE method:G₁,G₂are defined in section(2.3).AGE method of the two -dimensional equation is absolutely stable,too.Finally,through programs for the AGE methods for the one and two dimensionalproblems on parallel computers,we get the correct results in accord with the theorem.That is.when the scale of the problem is fixed,the speed-up ratio is be-coming bigger when the number of the processor increases.however the efficiency is decreasing.On the other side.when the number of the processor is fixed,the speed-up ratio and efficiency are becoming bigger when the scale of the problem increases.