Geodesic
Two-layer schemes of improved order of approximation for nonstationary problems in mathematical physics
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 50 (2010) no. 1, pp. 118-130 
P. N. Vabishchevich. Two-layer schemes of improved order of approximation for nonstationary problems in mathematical physics. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 50 (2010) no. 1, pp. 118-130. http://geodesic.mathdoc.fr/item/ZVMMF_2010_50_1_a9/
@article{ZVMMF_2010_50_1_a9,
     author = {P. N. Vabishchevich},
     title = {Two-layer schemes of improved order of approximation for nonstationary problems in mathematical physics},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {118--130},
     year = {2010},
     volume = {50},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2010_50_1_a9/}
}
TY  - JOUR
AU  - P. N. Vabishchevich
TI  - Two-layer schemes of improved order of approximation for nonstationary problems in mathematical physics
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2010
SP  - 118
EP  - 130
VL  - 50
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2010_50_1_a9/
LA  - ru
ID  - ZVMMF_2010_50_1_a9
ER  - 
%0 Journal Article
%A P. N. Vabishchevich
%T Two-layer schemes of improved order of approximation for nonstationary problems in mathematical physics
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2010
%P 118-130
%V 50
%N 1
%U http://geodesic.mathdoc.fr/item/ZVMMF_2010_50_1_a9/
%G ru
%F ZVMMF_2010_50_1_a9

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

In the theory of finite difference schemes, the most complete results concerning the accuracy of approximate solutions are obtained for two- and three-level finite difference schemes that converge with the first and second order with respect to time. When the Cauchy problem is numerically solved for a system of ordinary differential equations, higher order methods are often used. Using a model problem for a parabolic equation as an example, general requirements for the selection of the finite difference approximation with respect to time are discussed. In addition to the unconditional stability requirements, extra performance criteria for finite difference schemes are presented and the concept of SM stability is introduced. Issues concerning the computational implementation of schemes having higher approximation orders are discussed. From the general point of view, various classes of finite difference schemes for time-dependent problems of mathematical physics are analyzed.

[1] Samarskii A. A., Teoriya raznostnykh skhem, Nauka, M., 1989 | MR

[2] Samarskii A. A., Gulin A. V., Ustoichivost raznostnykh skhem, Nauka, M., 1973 | Zbl

[3] Samarskii A. A., Matus P. P., Vabishchevich P. N., Difference schemes with operator factors, Hardbound, Kluwer Acad. Publ., Dordrecht, 2002 | MR | Zbl

[4] Samarskii A. A., Vabischevich P. N., Chislennye metody resheniya zadach konvektsii-diffuzii, Editorial URSS, M., 2004

[5] Hundsdorfer W., Verwer J., Numerical solution of time-dependent advection-diffusion-reaction equations, Springer, Berlin, 2003 | MR | Zbl

[6] Gustafsson B., High order difference methods for time dependent PDE, Springer, Berlin, 2008 | MR | Zbl

[7] Ascher U. M., Numerical methods for evolutionary differential equations, SIAM, Philadelphia, PA, 2008 | MR

[8] LeVeque R. J., Finite difference methods for ordinary and partial differential equations. Steady-state and time-dependent problems, SIAM, Philadelphia, PA, 2007 | MR

[9] Rakitskii Yu. V., Ustinov S. M., Chernorutskii N. G., Chislennye metody resheniya zhestkikh sistem, Nauka, M., 1979 | MR

[10] Hairer E., Wanner G., Solving ordinary differential equations, II, Springer, Berlin, 1996 | MR | Zbl

[11] Butcher J. C., Numerical methods for ordinary differential equations, Wiley, Hoboken, N.Y., 2008 | MR | Zbl

[12] Dekker K., Verwer J., Stability of Runge–Kutta methods for stiff nonlinear differential equations, North-Holland, Amsterdam–New York-Oxford, 1984 | MR | Zbl

[13] Gear C. W., Numerical initial value problems in ordinary differential equations, Prentice-Hall, Englewood Cliffs, NJ, 1971 | MR | Zbl

[14] Samarskii A. A., Vabishchevich P. N., Computational heat transfer, v. 1, Math. Modelling, Wiley, Chichester, 1995

[15] Ladyzhenskaya O. A., Kraevye zadachi matematicheskoi fiziki, Nauka, M., 1973 | MR

[16] Higham N. J., Functions of matrices. Theory and computation, SIAM, Philadelphia, PA, 2008 | MR

[17] Moler C., Van Loan C., “Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later”, SIAM Rev., 45:1 (2003), 3–49 | DOI | MR | Zbl

[18] van der Vorst H. A., Iterative Krylov methods for large linear systems, Cambridge Univ. Press, Cambridge, 2003 | MR

[19] Saff E., Schonhage A., Varga R., “Geometric convergence to $e^{-z}$ by rational functions with real poles”, Numer. Math., 25 (1976), 307–322 | DOI | MR | Zbl

[20] Erlangga Y. A., “Advances in iterative methods and preconditioners for the Helmholtz equation”, Arch. Comput. Methods Engng., 15:1 (2008), 37–66 | DOI | MR | Zbl

[21] Axelsson O., Kucherov A., “Real valued iterative methods for solving complex symmetric linear systems”, Numer. Linear Algebra Appl., 7:4 (2000), 197–218 | 3.0.CO;2-S class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[22] Samarskii A. A., Nikolaev E. S., Metody resheniya setochnykh uravnenii, Nauka, M., 1978 | MR