The problem of parallel solution of partial differential equations with the help of the method of lines that ensures the reduction of the initial problem to the Cauchy problem described by a system of ordinary differential equations is considered. As a basic method, collocated multi-step block difference schemes are proposed. Obtaining a numerical solution in this case becomes possible only with the use of high-performance computing, usually with parallel architecture. This problem becomes especially important in the implementation of mathematical models based on systems of partial differential equations when it comes to the need for discretization of the solution search area, which can be significantly complicated by the geometric configuration of the boundaries. Solution of systems of this order cannot be obtained without involving multiprocessor computers. But by simply increasing the processing power, this problem cannot be solved. Only by combining the advantages of supercomputers and modern numerical simulation methods we can expect a significant improvement in the numerical solution of partial differential equations (PDEs).
The article is devoted to the development and justification of parallel numerical simulation methods. Results can be obtained to calculate the number of points. To increase the order of approximation of the developed methods, the set of points of the reference block (s) is introduced into difference schemes. Generation of difference schemes for any reference and calculated points is considered. For some block sizes, generated differential calculation schemes are obtained, oriented to the number of available processes in a parallel implementation. The absolute stability of the constructed multi-step multipoint methods according to the initial conditions and the right side is proved.
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