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Convergence of method of lines approximations to partial differential equations. (English) Zbl 0546.65064

Many existing numerical schemes for evolutionary problems in partial differential equations can be viewed as method of lines (MOL) schemes. This paper treats the convergence of one-step MOL schemes. Our main purpose is to set up a general framework for a convergence analysis applicable to nonlinear problems. The stability materials for this framework are taken from the field of nonlinear stiff ODEs. In this connection, important concepts are the logarithmic matrix norm and C- stability. A nonlinear parabolic equation and the cubic Schrödinger equation are used for illustrating the ideas.

MSC:

65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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[1] Axelsson, O.: Error estimates over infinite intervals of some discretizations of evolution equations. Technical Report, Catholic University of Nijmegen, 1983 (to appear in BIT).
[2] Burrage, K., Butcher, J. C.: Stability criteria for implicit Runge-Kutta methods. SIAM J. Numer. Anal.16, 46–57 (1979). · Zbl 0396.65043
[3] Dahlquist, G.: Stability and error bounds in the numerical integration of ordinary differential equations. Trans. Royal Inst. Techn., No 130, Stockholm, 1959. · Zbl 0085.33401
[4] Dekker, K., Verwer, J. G.: Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations. Amsterdam-New York-Oxford: North-Holland 1984. · Zbl 0571.65057
[5] Frank, R., Schneid, J., Ueberhuber, C. W.: The concept of B-convergence. SIAM J. Numer. Anal.18, 753–780 (1981). · Zbl 0467.65032
[6] Griffiths, D. F., Mitchell, A. R., Morris, J. LI: A numerical study of the nonlinear Schrödinger equations. Comp. Meth. Appl. Mech. Engn.45, 177–215 (1984). · Zbl 0555.65060
[7] Kreiss, H. O.: Über die Stabilitätsdefinition für Differenzengleichungen die partielle Differentialgleichungen approximieren. BIT2, 153–181 (1962). · Zbl 0109.34702
[8] Richtmyer, R. D., Morton, K. W.: Difference Methods for Initial Value Problems. New York: Interscience. 1967. · Zbl 0155.47502
[9] Sanz-Serna, J. M.: Convergent approximations to PDEs and contractivity of methods for stiff systems of ODEs. In: Actas del VI CEDYA, Jaca, 1983, pp. 488–493 (this paper is a highly condensed version of a report which is available on request).
[10] Sanz-Serna, J. M.: Methods for the numerical solution of the nonlinear Schrödinger equation. Math. of Comput.43, 21–24 (1984). · Zbl 0555.65061
[11] Verwer, J. G., Dekker, K.: Step-by-step stability in the numerical solution of partial differential equations. Report NW 161/83, Mathematical Centre, Amsterdam, 1983. · Zbl 0512.65054
[12] Verwer, J. G.: Contractivity of locally one-dimensional splitting methods. Numer. Math.44, 247–259 (1984). · Zbl 0539.65082
[13] Wirz, H. J.: On iterative solution methods for systems of partial differential equations. Lecture Notes in Mathematics, Vol. 679, pp. 151–163. Berlin-Heidelberg-New York: Springer 1978. · Zbl 0387.65056
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