Geodesic
Compact approximation of a two-dimensional boundary value problem for elliptic equations of the 2nd order with a discontinuous coefficient
Matematičeskoe modelirovanie, Tome 35 (2023) no. 4, pp. 88-119.

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For an elliptic equation of the 2nd order with variable discontinuous coefficients and the right side, a scheme of the 4th order of accuracy is constructed. On the jump line, the conditions of docking (Kirchhoff) are assumed to be fulfilled. The use of Richardson extrapolation, as numerical experiments have shown, increases the order of accuracy to about the 6th. It is shown that relaxation methods, including multigrid methods, are applicable to solving such systems linear algebraic equations (SLAE) corresponding to a compact finite-difference approximation of the problem. In comparison with the classical approximation, the accuracy increases by about 100 times with the same labor intensity. Various variants of the equation and boundary conditions are considered, as well as the problem of determining eigenvalues and functions for a piecewise constant coefficient of the equation.
Keywords: finite-difference approximation, stencil, test functions, accuracy order, Richardson extrapolation.
Mots-clés : compact implicit scheme 
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V. A. Gordin; D. A. Shadrin. Compact approximation of a two-dimensional boundary value problem for elliptic equations of the 2nd order with a discontinuous coefficient. Matematičeskoe modelirovanie, Tome 35 (2023) no. 4, pp. 88-119. http://geodesic.mathdoc.fr/item/MM_2023_35_4_a4/

[1] V. T. Zhukov, N. D. Novikova, O. B. Feodoritova, “Multigrid method for elliptic equations with anisotropic discontinuous coefficients”, Comput. Math. Math. Phys., 55 (2015), 1150–1163

[2] V. M. Babich, “Skin-effect v sluchae provoda proizvolnogo poperechnogo sechenia”, Zap. Nauchn. Sem. LOMI, 128 (1982), 13–20

[3] O. Ladyzhenskaya, N. Uraltseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York-London, 1968

[4] Sh. E. Mikeladze, “O chislennom integrirovanii uravnenii ellipticheskogo i parapolicheskogo tipov”, Izv. AN SSSR. Ser. Matem., 5:1 (1941), 57–74

[5] P. H. Cowell, A. C.D. Crommelin, Investigation of the motion of Halley's comet from 1759 to 1910. Appendix to Greenwich Observations for 1909, Edinburgh, 1910, 1–84

[6] B. V. Noumerov, “A Method of Extrapolation of Perturbations”, Monthly Notices Royal Astronomical Society, 84 (1924), 592–601

[7] E. Hairer, S. P. Norsett, G. Wanner, Solving ordinary differential equations, v. I, Nonstiff problems, Springer Verlag, New York, 1987, 1993, 480 pp.

[8] V. A. Gordin, E. A. Tsymbalov, “Kompaktnaia raznostnaia skhema dlia differentsialnogo uravneniia s kusochno-postoiannym koeffitsientom”, Matematicheskoe modelirovanie, 29:12 (2017), 16–28

[9] V. A. Gordin, Kak eto poschitat?, MTsNMO, M., 2005, 280 pp.

[10] M. M. Krasnov, “Parallel algorithm for computing points on a computation front hyperplane”, Comput. Math. and Math. Phys., 55 (2015), 140–147

[11] A. A. Samarskii, E. S. Nikolaev, Numerical Methods for Grid Equations, v. II, Iterative Methods, Birkhauser Verlag, Basel-Boston-Berlin, 1989

[12] R. P. Fedorenko, “A relaxation method for solving elliptic difference equations”, USSR Computational Mathematics and Mathematical Physics, 1:4 (1962), 1092–1096

[13] R. P. Fedorenko, Vvedenie v vychislitelnuiu fiziku, Intellekt, M.–Dolgoprudnyi, 2008

[14] A. Iserles, A First Course in the Numerical Analysis of Differential Equations, Cambridge University Press, 2009

[15] V. T. Zhukov, O. B. Feodoritova, “O razvitii parallelnyh algorithmov reshenia parabolicheskih i ellipticheskih uravnenii”, Itogi nauki i techniki. Seria sovremennaia matematika i ee prilojenia. Tematicheskii obzor, 155, 2018, 20–37