Geodesic
Numerical method for solving an initial-boundary value problem for a multidimensional loaded parabolic equation of a general form with conditions of the third kind
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 26 (2022) no. 1, pp. 7-35.

Voir la notice de l'article provenant de la source Math-Net.Ru

An initial-boundary value problem is studied for a multidimensional loaded parabolic equation of general form with boundary conditions of the third kind. For a numerical solution, a locally one-dimensional difference scheme by A.A. Samarskii with order of approximation $O(h^2+\tau)$ is constructed. Using the method of energy inequalities, we obtain a priori estimates in the differential and difference interpretations, which imply uniqueness, stability, and convergence of the solution of the locally one-dimensional difference scheme to the solution of the original differential problem in the $L_2$ norm at a rate equal to the order of approximation of the difference scheme. An algorithm for the computational solution is constructed and numerical calculations of test cases are carried out, illustrating the theoretical calculations obtained in this work.
Mots-clés : parabolic equation, convergence
Keywords: loaded equation, difference schemes, locally one-dimensional scheme, a priori estimate, stability, multidimensional problem. 
@article{VSGTU_2022_26_1_a0,
     author = {Z. V. Beshtokova},
     title = {Numerical method for solving an initial-boundary value problem for a multidimensional loaded parabolic equation of a general form with conditions of the third kind},
     journal = {Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences},
     pages = {7--35},
     publisher = {mathdoc},
     volume = {26},
     number = {1},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VSGTU_2022_26_1_a0/}
}
TY  - JOUR
AU  - Z. V. Beshtokova
TI  - Numerical method for solving an initial-boundary value problem for a multidimensional loaded parabolic equation of a general form with conditions of the third kind
JO  - Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences
PY  - 2022
SP  - 7
EP  - 35
VL  - 26
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/VSGTU_2022_26_1_a0/
LA  - ru
ID  - VSGTU_2022_26_1_a0
ER  - 
%0 Journal Article
%A Z. V. Beshtokova
%T Numerical method for solving an initial-boundary value problem for a multidimensional loaded parabolic equation of a general form with conditions of the third kind
%J Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences
%D 2022
%P 7-35
%V 26
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/VSGTU_2022_26_1_a0/
%G ru
%F VSGTU_2022_26_1_a0
Z. V. Beshtokova. Numerical method for solving an initial-boundary value problem for a multidimensional loaded parabolic equation of a general form with conditions of the third kind. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 26 (2022) no. 1, pp. 7-35. http://geodesic.mathdoc.fr/item/VSGTU_2022_26_1_a0/

[1] Nakhushev A. M., “Loaded equations and their applications”, Differ. Uravn., 19:1 (1983), 86–94 (In Russian) | MR

[2] Dyakonov E. G., “Difference schemes with a splitting operator for nonstationary equations”, Sov. Math., Dokl., 3:1 (1962), 645–648 | MR | Zbl

[3] Samarskii A. A., “On an economical difference method for the solution of a multidimensional parabolic equation in an arbitrary region”, USSR Comput. Math. Math. Phys., 2:5 (1963), 894–926 | DOI | MR | Zbl

[4] Samarskii A. A., “Local one dimensional difference schemes on non-uniform nets”, USSR Comput. Math. Math. Phys., 3:3 (1963), 572–619 | DOI | MR | Zbl

[5] Budak V. M., Iskenderov A. D., “A class of inverse boundary value problems with unknown coefficients”, Sov. Math., Dokl., 8:1 (1967), 1026–1030 | MR | Zbl

[6] Krall A. M., “The development of general differential and general differential boundary systems”, Rocky Mountain J. Math., 5:4 (1975), 493–542 | DOI

[7] Ladyzhenskaya O. A., The Boundary Value Problems of Mathematical Physics, Applied Mathematical Sciences, 49, Springer, New York, 1985, xxx+322 pp. | DOI

[8] Samarskii A. A., Teoriia raznostnykh skhem [Theory of Difference Schemes], Nauka, Moscow, 1983, 616 pp. (In Russian)

[9] Andreev V. B., “On the convergence of difference schemes approximating the second and third boundary value problems for elliptic equations”, USSR Comput. Math. Math. Phys., 8:6 (1968), 44–62 | DOI | MR | Zbl

[10] Samarskii A. A., Gulin A. B., Ustoichivost' raznostnykh skhem [Stability of Difference Schemes], Nauka, Moscow, 1973, 415 pp. (In Russian)

[11] Voevodin A. F., Shugrin S. M., Chislennye metody rascheta odnomernykh sistem [Numerical Methods for Calculating One-Dimensional Systems], Nauka, Novosibirsk, 1981, 208 pp. (In Russian)