On a new class of additive (splitting) operator-difference schemes

Vabishchevich, Petr

doi:10.1090/S0025-5718-2011-02492-0

On a new class of additive (splitting) operator-difference schemes
HTML articles powered by AMS MathViewer

by Petr N. Vabishchevich;

Math. Comp. 81 (2012), 267-276

DOI: https://doi.org/10.1090/S0025-5718-2011-02492-0

Published electronically: June 20, 2011

PDF | Request permission

Abstract:

Many applied time-dependent problems are characterized by an additive representation of the problem operator. Additive schemes are constructed using such a splitting and are associated with the transition to a new time level on the basis of the solution of more simple problems for the individual operators in the additive decomposition. We consider a new class of additive schemes for problems with additive representation of the operator at the time derivative. In this paper we construct and study the vector operator-difference schemes, which are characterized by a transition from the single initial evolution equation to a system of evolution equations.

References

V. N. Abrashin, A variant of the method of variable directions for the solution of multidimensional problems in mathematical physics. I, Differentsial′nye Uravneniya 26 (1990), no. 2, 314–323, 366 (Russian); English transl., Differential Equations 26 (1990), no. 2, 243–250. MR 1050397
Jim Douglas Jr. and H. H. Rachford Jr., On the numerical solution of heat conduction problems in two and three space variables, Trans. Amer. Math. Soc. 82 (1956), 421–439. MR 84194, DOI 10.1090/S0002-9947-1956-0084194-4
Richard E. Ewing, Numerical solution of Sobolev partial differential equations, SIAM J. Numer. Anal. 12 (1975), 345–363. MR 395265, DOI 10.1137/0712028
Richard E. Ewing, Time-stepping Galerkin methods for nonlinear Sobolev partial differential equations, SIAM J. Numer. Anal. 15 (1978), no. 6, 1125–1150. MR 512687, DOI 10.1137/0715075
D. G. Gordeziani and G. V. Meladze, The simulation of the third boundary value problem for multidimensional parabolic equations in an arbitrary domain by one-dimensional equations, Ž. Vyčisl. Mat i Mat. Fiz. 14 (1974), 246–250, 271 (Russian). MR 400734
Tarek P. A. Mathew, Domain decomposition methods for the numerical solution of partial differential equations, Lecture Notes in Computational Science and Engineering, vol. 61, Springer-Verlag, Berlin, 2008. MR 2445659, DOI 10.1007/978-3-540-77209-5
G. I. Marchuk, Splitting and alternating direction methods, Handbook of numerical analysis, Vol. I, Handb. Numer. Anal., I, North-Holland, Amsterdam, 1990, pp. 197–462. MR 1039325
D. W. Peaceman and H. H. Rachford Jr., The numerical solution of parabolic and elliptic differential equations, J. Soc. Indust. Appl. Math. 3 (1955), 28–41. MR 71874
A.A. Samarskii, Regularization of difference schemes, Computational Mathematics and Mathematical Physics 7 (1967), no. 1, 79–120.
Alexander A. Samarskii, The theory of difference schemes, Monographs and Textbooks in Pure and Applied Mathematics, vol. 240, Marcel Dekker, Inc., New York, 2001. MR 1818323, DOI 10.1201/9780203908518
A. A. Samarskiĭ and A. V. Gulin, Ustoĭchivost′raznostnykh skhem, 2nd ed., Èditorial URSS, Moscow, 2004 (Russian, with Russian summary). MR 2263771
A. A. Samarskii, P. P. Matus, and P. N. Vabishchevich, Difference schemes with operator factors, Mathematics and its Applications, vol. 546, Kluwer Academic Publishers, Dordrecht, 2002. MR 1950844, DOI 10.1007/978-94-015-9874-3
A.A. Samarskii, P.P. Matus, and P.N. Vabishchevich, Stability of vector additive schemes, Doklady Mathematics 58 (1998), no. 1, 133–135.
A. A. Samarskiĭ, Petr N. Vabishchevich, and Lubin G. Vulkov (eds.), Finite difference methods: theory and applications, Nova Science Publishers, Inc., Commack, NY, 1999. MR 1714030
A.A. Samarskii and P.N. Vabishchevich, Regularized additive full approximation schemes, Doklady Mathematics 57 (1998), 83–86.
S.L. Sobolev, Some new problems in mathematical physics, Izv. Akad. Nauk SSSR Ser. Mat 18 (1954), 3–50.
R. E. Showalter, Partial differential equations of Sobolev-Galpern type, Pacific J. Math. 31 (1969), 787–793. MR 252870
R. E. Showalter and T. W. Ting, Pseudoparabolic partial differential equations, SIAM J. Math. Anal. 1 (1970), 1–26. MR 437936, DOI 10.1137/0501001
A.G. Sveshnikov, A.B. Al’shin, M.O. Korpusov, and Y.D. Pletner, Linear and nonlinear equations of Sobolev type, Moscow: Fizmatlit. 734 p., 2007 (Russian).
P. N. Vabishchevich, Vector additive difference schemes for first-order evolution equations, Zh. Vychisl. Mat. i Mat. Fiz. 36 (1996), no. 3, 44–51 (Russian, with Russian summary); English transl., Comput. Math. Math. Phys. 36 (1996), no. 3, 317–322. MR 1382641
P. N. Vabishchevich, Regularized additive operator-difference schemes, Zh. Vychisl. Mat. Mat. Fiz. 50 (2010), no. 3, 449–457 (Russian, with Russian summary); English transl., Comput. Math. Math. Phys. 50 (2010), no. 3, 428–436. MR 2681922, DOI 10.1134/S096554251003005X
P. N. Vabishchevich, Domain decomposition methods with overlapping subdomains for the time-dependent problems of mathematical physics, Comput. Methods Appl. Math. 8 (2008), no. 4, 393–405. MR 2604752, DOI 10.2478/cmam-2008-0029
N. N. Yanenko, The method of fractional steps. The solution of problems of mathematical physics in several variables, Springer-Verlag, New York-Heidelberg, 1971. Translated from the Russian by T. Cheron. English translation edited by M. Holt. MR 307493