MUSCL reconstruction and Haar wavelets

MUSCL extensions (Monotone Upstream-centered Schemes for Conservation Laws) of the Godunov numerical scheme for scalar conservation laws are shown to admit a rather simple reformulation when recast in the formalism of the Haar multi-resolution analysis of $L^2(\mathbb{R})$. By pursuing this wavelet reformulation, a seemingly new MUSCL-WB scheme is derived for advection-reaction equations which is stable for a Courant number up to $1$ (instead of roughly $\frac{1}{2}$ ). However these high-order reconstructions aren’t likely to improve the handling of delicate nonlinear wave interactions in the involved case of systems of Conservation/Balance laws.

Keywords

Godunov scheme, Haar wavelets, multi-resolution analysis, MUSCL reconstruction, second-order resolution (SOR), slope-limiter, wave interactions, well-balanced (WB) scheme

2010 Mathematics Subject Classification

35Q35, 65M06, 65T60

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Published 13 May 2015