An adaptive wavelet-collocation method for shock computations

Abstract

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 Φ. 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 ϵ, 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.

Keywords:

• Euler equations
• hyperbolic conservation laws
• wavelets
• adaptive grid
• artificial viscosity
• Riemann problem
• shock capturing

Acknowledgements

Support for this work was provided by the National Science Foundation (NSF) under grants No. ACI-0242457 and CBET-0756046. This support is gratefully acknowledged.