Институт вычислительной математики
и математической геофизики

Abstracts

Numerical solution of differential and integral equations

In the report the results of studies of a method of a numerical solution of boundary value problems in inhomogeneous domains composed from homogeneous many-dimensional parallelepipeds are stated. The method represents symbiosis of the compact difference schemes and one-dimensional boundary conditions of any order. Inside homogeneous subdomains the high-order compact schemes are used, and on boundaries of the contact of mediums and on exterior boundaries the multipoint relations constructed on the basis of one-sided approximations of streams are used.

As a result of decomposition the many-dimensional boundary value problem is reduced to systems of the linear algebraic equations with matrixes of almost three-diagonal structure. Their difference from three-diagonal matrixes is, that the separate lines have more than three nonzero elements located near to a main diagonal. These lines correspond either exterior boundary conditions or conditions for streams on boundaries of contact of mediums.

For a solution of such systems of equationses two algorithms are offered. First is based on immediate transformation of separate fragments of a system to the three-diagonal form by a method of Gauss. The transformed complete system has three-diagonal structure. The second algorithm is based on a parallelizing of calculation on homogeneous subdomains. It represents generalization of a known method of a parallelizing for problems with a three-diagonal matrix. For both algorithms the sufficient conditions of a stability are established by research of a diagonal dominance.

Note. Abstracts are published in author's edition

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