An Adaptive Wavelet-Collocation Method for Shock Computations

A simple and robust method for solving hyperbolic conservation equations based on the adaptive wavelet-collocation method, which uses a dynamically adaptive grid, are presented. The method utilises natural ability of wavelet analysis to sense localised structures and is based on analysis of wavelet coefficients on the finest level of resolution to create a discontinuity locator function phi. Using this function, an artificial viscous term is explicitly added in the needed regions using a localised numerical viscosity that ensures the positivity and TVD non-linear stability conditions. Once the wavelet coefficients on the finest level of resolution are below the error threshold parameter epsilon, the artificial viscosity is shut off and any remaining physical waves are free to propagate undamped. Multiple examples in one and two dimensions are presented to demonstrate the method's robustness, simplicity and ease of extending to more complex problems.