'Les fleurs du mal' II: a dynamically adaptive wavelet method of arbitrary lines for nonlinear evolutionary problems-capturing steep moving fronts

Abstract

C. Baudelaire's 'les fleurs du mal' is an allusion to various new developments ('les fleurs') of the method of arbitrary lines (mal) [L.S. Xanthis, C. Schwab, The method of arbitrary lines, C.R. Acad. Sci. Paris, Sér. I 312 (1991) 181–187]. Here we extend the wavelet-mal methodology (C.R. Mécanique 362, 2004) to the solution of nonlinear evolutionary partial differential equations (PDE) in arbitrary domains, exemplified by Burgers’ equation. We employ the 'arbitrary Lagrangian–Eulerian' (ALE) formulation and some attractive properties of the wavelet approximation theory to develop a dynamically adaptive, wavelet-mal solver that is capable of capturing the anisotropic, or multi-scale character of the steep (shock-like) moving fronts that arise in such problems. We show the efficacy and high accuracy of the wavelet-mal methodology by numerical examples involving the Burgers' equation in two spatial dimensions.