Geodesic
Conditional space-time stability of collocation Runge-Kutta for parabolic evolution equations
Electronic transactions on numerical analysis, Tome 41 (2014), pp. 62-80
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.
Classification : 35K90, 65M12, 65M20, 65M60
Keywords: space-time variational formulation, Runge-Kutta collocation method, parabolic evolution equations, space-time stability, Petrov-Galerkin 
@article{ETNA_2014__41__a21,
     author = {Andreev,  Roman and Schweitzer,  Julia},
     title = {Conditional space-time stability of collocation {Runge-Kutta} for parabolic evolution equations},
     journal = {Electronic transactions on numerical analysis},
     pages = {62--80},
     year = {2014},
     volume = {41},
     zbl = {1297.65104},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_2014__41__a21/}
}
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AU  - Schweitzer,  Julia
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EP  - 80
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UR  - http://geodesic.mathdoc.fr/item/ETNA_2014__41__a21/
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%A Schweitzer,  Julia
%T Conditional space-time stability of collocation Runge-Kutta for parabolic evolution equations
%J Electronic transactions on numerical analysis
%D 2014
%P 62-80
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%U http://geodesic.mathdoc.fr/item/ETNA_2014__41__a21/
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%F ETNA_2014__41__a21
Andreev,  Roman; Schweitzer,  Julia. Conditional space-time stability of collocation Runge-Kutta for parabolic evolution equations. Electronic transactions on numerical analysis, Tome 41 (2014), pp. 62-80. http://geodesic.mathdoc.fr/item/ETNA_2014__41__a21/