High-order second-generation wavelet upwind schemes with multiresolution self-adaptive capabilities for hyperbolic conservation laws

We introduce a novel system of high-order wavelet collocation upwind schemes utilizing a broad class of second-generation wavelets for hyperbolic conservation laws. These schemes possess multiresolution self-adaptive capabilities and are integrated into a meshfree framework. Unlike traditional high-order schemes that necessitate frequent variable transformations between physical and characteristic spaces involving numerous local projections, our schemes enable the direct discretization of relevant spatial derivatives in the physical space. The newly proposed schemes make the most use of wavelet properties, i.e., applying asymmetrical interpolating wavelets to achieve the upwind property and utilizing the symmetrical second-generation wavelets to preserve shapes of functions and obtain higher data compression rates. Numerical tests of which solutions contain strong discontinuities and different scale smooth structures are conducted to demonstrate the enhanced performance of the proposed schemes based on the second-generation wavelets. The proposed schemes can utilize only 9 % of the nodes in the WENO5 scheme to yield similar results when addressing the advection of a square wave and achieve the convergent solution for the two interacting blast waves problem without local characteristics projection. In comparison to conventional methods and wavelet upwind schemes based on the interpolating wavelets we previously introduced, the newly proposed schemes exhibit a significantly higher data compression rate and yield substantial computational savings without compromising accuracy.