High-order adaptive multiresolution wavelet upwind schemes for hyperbolic conservation laws

We present a novel system of high-order adaptive multiresolution wavelet collocation upwind schemes, implemented in a meshfree framework, for solving hyperbolic conservation laws. Our approach constructs asymmetrical wavelet bases with interpolation properties to achieve the upwind property and address nonlinearities in hyperbolic problems. An adaptive algorithm based on multiresolution analysis in wavelet theory is designed to capture moving shock waves and identify new localized steep regions. To suppress the Gibbs phenomenon, we propose an integration average reconstruction method based on the Lebesgue differentiation theorem. These numerical techniques enable the wavelet collocation upwind scheme to provide a general framework for devising satisfactory adaptive wavelet upwind methods with high-order accuracy. We perform several benchmark tests for 1D hyperbolic problems to verify the accuracy and efficiency of the proposed wavelet schemes. Notably, our wavelet collocation upwind schemes successfully solve the two interacting blast waves problem when discretizing the spatial derivatives in the physical space directly. This is in contrast to classical high-order schemes that usually involve transforming variables back and forth between physical and characteristic spaces with many local projections. Our results indicate the robustness, stability, and efficiency of our proposed schemes for strong shock wave interaction problems.