Convergence of Runge-Kutta methods for nonlinear parabolic equations

Authors:
Alexander Ostermann;Mechthild Thalhammer
Affiliations:
Institut für Technische Mathematik, Geometrie und Bauinformatik, Universität Innsbruck, Technikerstrasse 13, A-6020 Innsbruck, Austria;Institut für Technische Mathematik, Geometrie und Bauinformatik, Universität Innsbruck, Technikerstrasse 13, A-6020 Innsbruck, Austria
Venue:
Applied Numerical Mathematics
Year:
2002

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Abstract

In this paper, we study time discretizations of fully nonlinear parabolic differential equations. Our analysis uses the fact that the linearization along the exact solution is a uniformly sectorial operator. We derive smooth and nonsmooth-data error estimates for the backward Euler method, and we prove convergence for strongly A(θ)-stable Runge-Kutta methods. For the latter, the order of convergence for smooth solutions is essentially determined by the stage order of the method. Numerical examples illustrating the convergence estimates are presented.