Adaptive multiresolution methods

¹ Laboratório Associado de Computação e Matemática Aplicada (LAC), Coordenadoria dos Laboratórios Associados (CTE), Instituto Nacional de Pesquisas Espaciais (INPE), Av. dos Astronautas 1758, 12227-010 São José dos Campos, São Paulo, Brazil
(e-mail: margarete@lac.inpe.br)
² Instituto de Matemática, Estatística e Computação Científica (IMECC), Universidade Estadual de Campinas (Unicamp) , R. Sérgio Buarque de Holanda 651, 13083-970 Campinas, São Paulo, Brazil
(e-mail: soniag@ime.unicamp.br)
³ Centre de Mathématiques et Leurs Applications (CMLA), Ecole Normale Supérieure de Cachan, 61 avenue du President Wilson, 94235 Cachan cedex, France
(e-mail: o_roussel@yahoo.fr)
⁴ Laboratoire de Mécanique, Modelisation et Procédés Propres (M2P2), CNRS, and Centre de Mathétiques e d’Informatique (CMI), Université de Provence, 39 rue F. Joliot-Curie, 13451 Marseille Cedex 13, France
(e-mail: kschneid@cmi.univ-mrs.fr)

Abstract

These lecture notes present adaptive multiresolution schemes for evolutionary PDEs in Cartesian geometries. The discretization schemes are based either on finite volume or finite difference schemes. The concept of multiresolution analyses, including Harten’s approach for point and cell averages, is described in some detail. Then the sparse point representation method is discussed. Different strategies for adaptive time-stepping, like local scale dependent time stepping and time step control, are presented. Numerous numerical examples in one, two and three space dimensions validate the adaptive schemes and illustrate the accuracy and the gain in computational efficiency in terms of CPU time and memory requirements. Another aspect, modeling of turbulent flows using multiresolution decompositions, the so-called Coherent Vortex Simulation approach is also described and examples are given for computations of three-dimensional weakly compressible mixing layers. Most of the material concerning applications to PDEs is assembled and adapted from previous publications [27, 31, 32, 34, 67, 69].