, Volume 13, Issue 2, pp 187-220
Date: 07 Nov 2012

Convergence of Lowest Order Semi-Lagrangian Schemes

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Abstract

We consider generalized linear transient advection-diffusion problems for differential forms on a bounded domain in ℝ d . We provide comprehensive a priori convergence estimates for their spatiotemporal discretization by means of a first-order in time semi-Lagrangian approach combined with a discontinuous Galerkin method. Under rather weak assumptions on the velocity underlying the advection we establish an asymptotic L 2-estimate of order \(O(\tau+h^{r}+h^{r+1}\tau^{-\frac{1}{2}}+\tau^{\frac{1}{2}})\) , where h is the spatial meshwidth, τ denotes the time step, and r is the polynomial degree of the forms used as trial functions. This estimate can be improved considerably in a variety of special settings.

Communicated by Douglas Arnold.