How to predict accurate wavelet grids in adaptive semi-Lagrangian schemes?

Martin Campos Pinto¹

¹ Institut de Recherche Mathématique Avancée (IRMA), CNRS & Université de Strasbourg, 67084 Strasbourg, France

ESAIM. Proceedings, Tome 29 (2009), pp. 43-57.

Résumé

In this article I present a new adaptive semi-Lagrangian scheme based on wavelet approximations for solving transport equations with underlying smooth flow. Inspired by the method of Besse, Filbet, Gutnic, Paun and Sonnendrücker [1], this new approach differs in the fact that it is mostly driven by the notion of good adaptation of a wavelet tree to a given function. Moreover it comes with guaranteed error estimates.
In a previous joint work with Mehrenberger [3], we had designed a first adaptive semi-Lagrangian scheme based on multilevel, hierarchical meshes. The method consisted in predicting a new adaptive mesh for every time step by using a low-cost strategy, and next readapt it once according to the smoothness of the transported numerical solutions. By a rigorous analysis we could prove that our scheme had a prescribed accuracy, achieved by applying the prediction and correction algorithms only once per time step.
The present scheme implements similar ideas, but now in the framework of interpolatory wavelets. For this purpose I translate the property of being (strongly) well-adapted to a given function in the context of wavelet trees, and show that it is (weakly) preserved by a low-cost prediction algorithm which transports wavelet grids along any smooth flow. As a consequence, error estimates can be established for the resulting “predict and readapt” scheme under the essential assumption that the flow underlying the transport equation, as well as its numerical approximation, is a stable diffeomorphism. One complexity result is stated in addition. The proofs can be found in the lecture notes [2].

Détail

DOI : 10.1051/proc/2009054