and Computational Methods

doi: 10.18698/2309-3684-2023-3-105124

The purpose of the work is to build and implement an algorithm for finding a numerical solution to a problem for mixed-type equations in an unlimited region. In this case problems are considered in which the process under study is described in some limited area by the thermal conductivity equation or wave equation, and outside it by the Laplace equation. The necessary additional conditions at zero, at infinity and the conditions for conjunction at the border of the inner region are set. There is described an algorithm for finding a numerical solution to a problem with a wave equation in a limited region in one-dimensional and two-dimensional cases, problems with a thermal conductivity equation or a wave equation in a two-dimensional case. Difference schemes are built by the integro–interpolation method. The task is solved in a limited area. Nonlocal boundary conditions are set on its border so the solution of task in limited area coincides with projection of problem in unlimited area. In this case, an artificial boundary is introduced for the solution in the part of the region in which the process is described by the Laplace equation. An iterative algorithm and an algorithm with a non-local boundary condition are built. The results of calculations for examples in various fields are presented.

Koleva M.N. Numerical solution of the heat equation in unbounded domains using quasi-uniform grids. Lecture Notes in Computer Science, 2006, vol. 3743, pp. 509–517.

Ryabenky V.S. Metod raznostnyh potencialov dlya nekotoryh zadach mekhaniki sploshnyh sred [The method of difference potentials for some problems of continuum mechanics]. Moscow, Nauka Publ., 1987, 391 p.

Brushlinskii, K.V., Ryaben'kii, V.S., Tuzova, N.B. The transfer of boundary conditions across a vacuum in axisymmetric problems. Computational Mathematics and Mathematical Physics, 1992, vol. 32, no. 12, pp. 1757–1767.

Brushlinsky K.V. Matematicheskie i vychislitel'nye zadachi magnitnoj gidrodinamiki [Mathematical and computational problems of magnetic hydrodynamics]. Moscow, BINOM. Laboratory of Knowledge Publ., 2009, 200 p.

Bettess P. Infinite Elements. Paris, Penshaw Press., 1992, 264 p.

Zienkiewicz O.C., Emson C., Bettess P. A novel boundary infinite element. International Journal for Numerical Methods in Engineering, 1983, vol. 83, no. 3, pp. 393–404.

Kalitkin N.N., Aleshin A.B., Alshina E.A., Rogov B.V. Vychisleniya na kvaziravnomernyh setkah [Calculations on quasi-uniform grids]. Moscow, Fizmatlit Publ., 2005, 223 p.

Galanin M.P., Nizkaya T.V. Development and application of a numerical method for solution of linear elliptic equations in unbounded region. Keldysh Institute Preprints, 2005, no. 2, pp. 1–29.

Tsynkov S.V. Numerical solution of problems on unbounded domains. A review. Applied Numerical Mathematics, 1998, vol. 27, iss. 4, pp. 465–532.

Galanin M.P., Sorokin D.L. Solving exterior boundary value problems for the Laplace equation. Differential Equations, 2020, vol. 56, no. 7, pp. 890–899.

Galanin M.P., Sorokin D.L., Ukhova A.R. Methods for numerical solution of a mixed type differential equation in an unbounded domain. Mathematical Modeling and Computational Methods, 2021, no. 1, с. 91–109.

Galanin MP, Sorokin DL, Ukhova AR. On solving the equation of mixed type in an unlimited region. Differential equations, 2022, vol. 58, no. 7, pp. 921–929.

Tikhonov A.N., Samarsky A.A. Uravneniya matematicheskoj fiziki [Equations of mathematical physics]. Moscow, Nauka Publ., 1972, 735 p.

Sveshnikov A.G., Bogolyubov A.N., Kravtsov V.V. Lekcii po matematicheskoj fizike [Lectures on mathematical physics]. Moscow, MSU Publ., 1993, 352 p.

Martinson L.K., Malov Yu.I. Differencial'nye uravneniya matematicheskoj fiziki [Differential equations of mathematical physics]. Moscow, BMSTU Publ., 1996, 228 p.

Galanin M.P., Savenkov E.B. Metody chislennogo analiza matematicheskih modelej [Methods of numerical analysis of mathematical models]. Moscow, BMSTU Publ., 2010, 591 p.

Samarskiy A.A. Vvedenie v teoriyu raznostnyh skhem [Introduction to the theory of difference schemes]. Moscow, Nauka Publ., 1971, 552 p.

Eigen is a C++ template library for linear algebra: matrices, vectors, numerical solvers, and related algorithms [Electronic resource]. URL: https://eigen.tuxfamily.org (accessed: 31.03.2022)

Galanin M.P., Sorokin D.L. Development and application of numerical methods for solving tasks in unlimited regions based on the third green formula. Keldysh Institute Preprints, 2018, no. 246, pp. 1–24.

Галанин М.П., Ухова А.Р. Численное решение уравнений смешанного типа в неограниченной области на плоскости. Математическое моделирование и численные методы, 2023, № 3, с. 105–124.

Исследование выполнено за счёт Российского научного фонда (грант 22-21-00260).

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