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Erschienen in:
2017 | OriginalPaper | Buchkapitel
Optimal Order of Convergence for the Upwind Scheme for the Linear Advection on a Bounded Domain
verfasst von : Nina Aguillon, Franck Boyer
Erschienen in: Finite Volumes for Complex Applications VIII - Methods and Theoretical Aspects
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Abstract
This proceeding presents an optimal error estimate in the \(L^{1}\)-norm of order 1 / 2 between the exact solution of an initial and boundary value problem for the linear advection equation and its approximation by the explicit upwind scheme. The space domain is bounded and a Dirichlet condition is thus imposed on the entering part of the boundary. This result extends the analysis given in Merlet and Vovelle (Numer. Math. 106(1), 129–155 (2007), [10]) that concerns the case where the equation is posed on the whole space. One of the key point of the proof is the analysis of a suitable regularization by convolution of the exact (weak) solution. Compared to Merlet and Vovelle (Numer. Math. 106(1), 129–155 (2007), [10]) we also relax some hypothesis on the velocity field, which in particular is allowed to be somehow discontinuous in time. This proceeding is a short version of Aguillon and Boyer (IMA J. Numer. Anal. (2017), [1]), aiming to present the steps of the proof and the new intermediate results.