Mathematics of Computation

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ISSN 1088-6842(e) ISSN 0025-5718(p)

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

Author(s): Etienne Emmrich; Mechthild Thalhammer.
Journal: Math. Comp. 79 (2010), 785-806.
MSC (2000): Primary 65M12, 65M15, 47J35, 35K55, 47H05
Posted: July 23, 2009
MathSciNet review: 2600543
Retrieve article in: PDF

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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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