Condition Analysis And Research Of Parallel Computation Of Equations

In general,it is difficult to obtain accurate solutions for partial differential equations,so the study of their numerical solutions has become very important.In order to reduce the error of numerical solutions of partial differential equations in calculations,accuracy often be improved by adding the number of discrete nodes.But it will increase the computational workload,computational difficulty,and waste the computational time.So we consider to use multiple computers to realize computation.Thus we carry on research on parallel algorithms.This article extends the parallel algorithms,which used to solve numerical solutions of partial differential equations in the past.We establish one method,which can construct parallelizable numerical scheme,improve the computational efficiency of numerical solution and widen the scope of parallel computation for algebraic equations.This method makes parallel algorithms to be more suitable for general situations.And on this basis,we use the generalized parallel algorithm to solve the numerical solution of Laplace equation.Firstly,considering that Lawrie Sameh parallel algorithm only applies to the case which coefficient matrix of linear equations is diagonal and sparse,this paper breaks this limitation and makes the parallel algorithm more general.With this goal,we divide coefficient matrix into blocks matrix,and then simplifies the linear equations through matrix sparsity.Thereby we achieve parallel computation for linear equations.In fact,there is a more general case,that is linear equations can’t realize parallel computation after coefficient matrix is divided into matrix blocks.On this basis,for this kind of case,we can construct invertible block matrix from original coefficient matrix,by searching its rows and columns,and then make pretreatment for coefficient matrix.Thus parallel numerical calculation is realized.Consequently,we promote Lawrie Sameh parallelizable algorithm and improve numerical computation efficiency.Secondly,by finite difference method,we take two-dimensional Laplace equation as an example.Based on this,a numerical scheme which can realize parallel computation is constructed.With Taylor formula,we construct a numerical format with second-order precision at the node,and the coefficient matrix of the constructed linear equations is partitioned and sparsely preprocessed.And then one sub-equations are extracted from original equations.A small number of unknown variables are calculated first,and on this basis,most of the remaining unknown variables are calculated in parallel later.So far,the parallelism of numerical format is verified theoretically.At the same time,numerical analysis is carried out to further verify the superiority and effectiveness of parallel scheme constructed.Finally,based on variable limit integral method,multiple variable limit integral is performed on the original differential equation at the node.Our goal is to eliminate the derivative in the original differential equation,so that it is transformed into an integral equation.Then we construct a parallel discrete scheme with fourth order precision.For the linear equations obtained,the coefficient matrix is processed in blocks first.On this basis,the universal parallel algorithm established in this paper is used to pre-process the coefficient matrix,that is to build reversible block matrix by searching rows and columns and sparse processing.Until now,one sub-equations are extracted from original equations.And a small number of unknown variables are solved first.Then these small number of unknown variables are substituted back into the original linear equations,and parallel calculation is carried out to realize the calculation for the remaining unknown variables.Thus,the computational complexity of time and space is reduced to a certain extent,and the computational accuracy and efficiency are improved.At the same time,numerical analysis is carried out to further verify the validity of the parallel scheme.