Adaptive-Anisotropic Wavelet Collocation Method on general curvilinear coordinate systems

Abstract

A new general framework for an Adaptive-Anisotropic Wavelet Collocation Method (A-AWCM) for the solution of partial differential equations is developed. This proposed framework addresses two major shortcomings of existing wavelet-based adaptive numerical methodologies, namely the reliance on a rectangular domain and the “curse of anisotropy”, i.e. drastic over-resolution of sheet- and filament-like features arising from the inability of the wavelet refinement mechanism to distinguish highly correlated directional information in the solution. The A-AWCM addresses both of these challenges by incorporating coordinate transforms into the Adaptive Wavelet Collocation Method for the solution of PDEs. The resulting integrated framework leverages the advantages of both the curvilinear anisotropic meshes and wavelet-based adaptive refinement in a complimentary fashion, resulting in greatly reduced cost of resolution for anisotropic features. The proposed Adaptive-Anisotropic Wavelet Collocation Method retains the a priori error control of the solution and fully automated mesh refinement, while offering new abilities through the flexible mesh geometry, including body-fitting. The new A-AWCM is demonstrated for a variety of cases, including parabolic diffusion, acoustic scattering, and unsteady external flow.