Ren, X. and Xanthis, L. 2006. 'Les fleurs du mal' II: a dynamically adaptive wavelet method of arbitrary lines for nonlinear evolutionary problems-capturing steep moving fronts. *Computer Methods in Applied Mechanics and Engineering.* 195 (37-40), pp. 4962-4970. doi:10.1016/j.cma.2005.10.022

Title | 'Les fleurs du mal' II: a dynamically adaptive wavelet method of arbitrary lines for nonlinear evolutionary problems-capturing steep moving fronts |
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Authors | Ren, X. and Xanthis, L. |

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. |

Keywords | Method of arbitrary lines, Spaceâtime ALE method, Nonlinear PDE system, Burgers' equation, Time-dependent convectionâdiffusion problem, Anisotropic discretization, Multi-scale wavelet approximation, Nonlinear ODE system |

Journal | Computer Methods in Applied Mechanics and Engineering |

Journal citation | 195 (37-40), pp. 4962-4970 |

ISSN | 0045-7825 |

Year | 15 Jul 2006 |

Digital Object Identifier (DOI) | doi:10.1016/j.cma.2005.10.022 |

Publication dates | |

Published | 15 Jul 2006 |