Multiresolution-based adaptive central high resolution schemes for modeling of nonlinear propagating fronts

Abstract

A class of wavelet-based methods, which combine adaptive grids together with second-order central and central-upwind high-resolution schemes, is introduced for accurate solution of first order hyperbolic systems of conservation laws and related equations. It is shown that the proposed class recovers stability which otherwise may be lost when the underlying central schemes are utilized over non-uniform grids. For simulations on irregular grids, both full-discretized and semi-discretized forms are derived; in particular, the effect of using certain shifts of the center of the computational allow to use standard slope limiters on non-uniform cells. Thereafter, the questions of numerical entropy production, local truncation errors, and the Total Variation Diminishing (TVD) criterion for scalar equations, are investigated. Central schemes are sensitives for irregular grids. To cure this drawback in the present context of multiresolution, the adapted grid is locally modified around high-gradient zones by adding new neighboring points. For an adapted point of resolution scale j, the new points are locally added in both resolution j and all successive coarser resolutions. It is shown that this simple grid modification improves stability of numerical solutions; this, however, comes at the expense that cell-centered central and central-upwind high resolution schemes, may not satisfy the TVD property. Finally, we present numerical simulations for both scalar and systems of non-linear conservation laws which confirm the simplicity and efficiency of the proposed method. These simulations demonstrate the high accuracy, the entropy production of wavelet-based algorithms which can effectively detect high gradient zones of shock waves, rarefaction regions, and contact discontinuities.