Runge-Kutta time discretizations of nonlinear dissipative evolution equations

Hansen, Eskil

doi:10.1090/S0025-5718-05-01866-1

Runge-Kutta time discretizations of nonlinear dissipative evolution equations
HTML articles powered by AMS MathViewer

by Eskil Hansen;

Math. Comp. 75 (2006), 631-640

DOI: https://doi.org/10.1090/S0025-5718-05-01866-1

Published electronically: December 19, 2005

PDF | Request permission

Abstract:

Global error bounds are derived for Runge-Kutta time discretizations of fully nonlinear evolution equations governed by $m$-dissipative vector fields on Hilbert spaces. In contrast to earlier studies, the analysis presented here is not based on linearization procedures, but on the fully nonlinear framework of logarithmic Lipschitz constants in order to extend the classical $B$-convergence theory to infinite-dimensional spaces. An algebraically stable Runge-Kutta method with stage order $q$ is derived to have a global error which is at least of order $q-1$ or $q$, depending on the monotonicity properties of the method.

References

V. Barbu, Nonlinear semigroups and differential equations in Banach spaces, Noordhoff International Publishing, Leyden, 1976.
M. G. Crandall, Nonlinear semigroups and evolution governed by accretive operators, Proc. Sympos. Pure Math. 45 (1986), pp. 305–337.
K. Dekker and J. G. Verwer, Stability of Runge-Kutta methods for stiff nonlinear differential equations, North-Holland Publishing Co., Amsterdam, 1984.
R. Frank, J. Schneid and C. W. Ueberhuber, Order results for implicit Runge-Kutta methods applied to stiff systems, SIAM J. Numer. Anal. 22 (1985), pp. 515–534.
C. González, A. Ostermann, C. Palencia and M. Thalhammer, Backward Euler discretization of fully nonlinear parabolic problems, Math. Comp. 71 (2002), pp. 125–145.
E. Hairer and G. Wanner, Solving ordinary differential equations. II. Stiff and differential-algebraic problems, Second edition, Springer-Verlag, Berlin, 1996.
E. Hansen, Convergence of multistep time discretizations of nonlinear dissipative evolution equations, To appear in SIAM J. Numer. Anal.
I. Higueras and G. Söderlind, Logarithmic norms and nonlinear DAE stability, BIT 42 (2002), pp. 823–841.
Ch. Lubich and A. Ostermann, Runge-Kutta methods for parabolic equations and convolution quadrature, Math. Comp. 60 (1993), pp. 105–131.
—, Runge-Kutta approximation of quasilinear parabolic equations, Math. Comp. 64 (1995), pp. 601–627.
A. Ostermann and M. Thalhammer, Convergence of Runge-Kutta methods for nonlinear parabolic equations, Appl. Numer. Math. 42 (2002), pp. 367–380.
A. Ostermann, M. Thalhammer and G. Kirlinger, Stability of linear multistep methods and applications to nonlinear parabolic problems, Appl. Numer. Math. 48 (2004), pp. 389–407.
A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York, 1983.
G. Söderlind, On nonlinear difference and differential equations, BIT 24 (1984), pp. 667–680.
—, Bounds on nonlinear operators in finite-dimensional Banach spaces, Numer. Math. 50 (1986), pp. 27–44.
V. Thomée, Galerkin finite element methods for parabolic problems, Springer-Verlag, New York, 1997.
E. Zeidler, Nonlinear functional analysis and its applications. II/B. Nonlinear monotone operators, Springer-Verlag, New York, 1990.