On solving systems of ordinary differential equations on MIMD-computers

Abstract

The error of solution of Cauchy problems for systems of ordinary differential equations is estimated in the case where the input data are approximate. It is shown how to prepare a program for computing the right-hand sides of the system automatically and simultaneously. Diagrams are presented to illustrate the efficiency of parallelization.Keywords: Cauchy problem for systems of ordinary differential equations, distributed-memory MIMD computers, parallelizing of computations, approximate initial data.A need often arises in various applications to solve a system of ordinary differential equations (SODE) with initial conditions (i.e., Cauchy problems for SODE). They may both be an independent class of problems and arise during solution of more complicated mathematical problems. Such problems have to be solved in, for example, describing time-dependent motions, processes, and phenomena. These systems describe, in particular, chemical kinetics and processes in nuclear reactors, dynamic problems of strength analysis of structures based on the three-dimensional finite-element method, etc.In problems concerned with the motion of controlled objects, SODE should often be solved faster than the process runs in real time, the more so that their solution requires multivariant computing. Such problems can be solved efficiently using computers that allow parallel computations, in particular, MIMD computers [1]. The present paper shows how to implement efficiently some basic algorithms on distributed-memory MIMD computers to solve systems of ordinary differential equations with approximate initial data.Let us consider a Cauchy problem for an SODE of the nth order on the interval [ , ] t T 0

Intelligent technologies of high-performance computing

“…In [6][7][8][9][10][11], questions of reliability of obtaining computer solutions to problems of various classes are considered.…”

“…The problem of reliability of computer solutions still remains one of practically important problems. An investigation of total errors is presented in [6][7][8][9][10][11].…”