Volume 41, pp. 62-80, 2014.

Conditional space-time stability of collocation Runge-Kutta for parabolic evolution equations

Roman Andreev and Julia Schweitzer

Abstract

We formulate collocation Runge–Kutta time-stepping schemes applied to linear parabolic evolution equations as space-time Petrov–Galerkin discretizations, and investigate their a priori stability for the parabolic space-time norms, that is the operator norm of the discrete solution mapping. The focus is on A-stable Gauß-Legendre and L-stable right-Radau nodes, addressing in particular the implicit midpoint rule, the backward Euler, and the three stage Radau5 time-stepping schemes. Collocation on Lobatto nodes is analyzed as a by-product. We find through explicit estimates that the operator norm is controlled in terms of the parabolic CFL number together with a measure of self-duality of the spatial discretization. Numerical observations motivate and illustrate the results.

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Key words

Space-time variational formulation, Runge-Kutta collocation method, parabolic evolution equations, space-time stability, Petrov-Galerkin.

AMS subject classifications

35K90, 65M12, 65M20, 65M60

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