Application Of Wavelet Analysis To Solving Hyperbolic Conservation Laws

The numerical computation for physical solutions of hyperbolic systems of conservation laws is difficult due to the presence of strong discontinuities in the solution. Uniformly high-order accurate ENO (Essentially Non-Oscillatory) schemes succeed in computing highly accurate numerical solutions to hyperbolic systems of conservation laws, typically second or three order in smooth regions, while maintaining sharp, oscillation free, numerical profiles at discontinuities, but usually their computational cost is quite large. Inthis paper, we construct a multi-resolution analysis of H2 (I) space using spline functionand give the multi-level decomposition of a function through interpolating spline wavelet transform. It is based on a point-value multi-resolution analysis that is used to detect regions with singularities (e.g. shocks), instead of a cell-average value multi-resolution analysis. In these regions, an expensive high-order accurate ENO scheme is applied to evaluate the numerical flux at cell boundaries. And hi smooth regions a cheap spline interpolation is used to get the value of the numerical divergence from values previously obtained on lower resolution scales to save the computational cost. Some other techniques are also used hi this method for improving its efficiency. At last this method is applied to 2-D hyperbolic conservation laws, and can be extended to higher dimensions easily for the computation of numerical divergence dimension by dimension.