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Stiffly accurate Runge–Kutta methods for nonlinear evolution problems governed by a monotone operator

Authors: Etienne Emmrich and Mechthild Thalhammer
Journal: Math. Comp. 79 (2010), 785-806
MSC (2000): Primary 65M12, 65M15, 47J35, 35K55, 47H05
Published electronically: July 23, 2009
MathSciNet review: 2600543
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Abstract: Stiffly accurate implicit Runge–Kutta methods are studied for the time discretisation of nonlinear first-order evolution equations. The equation is supposed to be governed by a time-dependent hemicontinuous operator that is (up to a shift) monotone and coercive, and fulfills a certain growth condition. It is proven that the piecewise constant as well as the piecewise linear interpolant of the time-discrete solution converges towards the exact weak solution, provided the Runge–Kutta method is consistent and satisfies a stability criterion that implies algebraic stability; examples are the Radau IIA and Lobatto IIIC methods. The convergence analysis is also extended to problems involving a strongly continuous perturbation of the monotone main part.

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Additional Information

Etienne Emmrich
Affiliation: Technische Universität Berlin, Institut für Mathematik, Straße des 17. Juni 136, 10623 Berlin, Germany

Mechthild Thalhammer
Affiliation: Leopold-Franzens-Universität, Institut für Mathematik, Technikerstraße 13/VII, 6020 Innsbruck, Austria
MR Author ID: 661917

Received by editor(s): September 19, 2008
Received by editor(s) in revised form: April 16, 2009
Published electronically: July 23, 2009
Article copyright: © Copyright 2009 American Mathematical Society