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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Stiffly accurate Runge–Kutta methods for nonlinear evolution problems governed by a monotone operator
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by Etienne Emmrich and Mechthild Thalhammer PDF
Math. Comp. 79 (2010), 785-806 Request permission

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
  • Email: emmrich@math.tu-berlin.de
  • Mechthild Thalhammer
  • Affiliation: Leopold-Franzens-Universität, Institut für Mathematik, Technikerstraße 13/VII, 6020 Innsbruck, Austria
  • MR Author ID: 661917
  • Email: Mechthild.Thalhammer@uibk.ac.at
  • Received by editor(s): September 19, 2008
  • Received by editor(s) in revised form: April 16, 2009
  • Published electronically: July 23, 2009
  • © Copyright 2009 American Mathematical Society
  • Journal: Math. Comp. 79 (2010), 785-806
  • MSC (2000): Primary 65M12, 65M15, 47J35, 35K55, 47H05
  • DOI: https://doi.org/10.1090/S0025-5718-09-02285-6
  • MathSciNet review: 2600543