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

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