Nonlinear anisotropic diffusion filters for the numerical approximation of conservation laws

This work integrates nonlinear discrete dissipation models -- namely anisotropic diffusion filters -- into the field of the numerical approximation for conservation laws is accomplished. Second-order accurate methods, namely the Lax-Wendroff scheme, are stabilised using the anisotropic dissipation filters combined with the shock detector. A new class of numerical schemes with nonlinear anisotropic dissipation models are derived. These algorithms nearly reduce all oscillations while the sharp shock resolution of the high-order scheme is maintained. Furthermore, emanating from the positivity requirement an alternative scheme is developed which is founded on the wave propagation algorithms developed by LeVeque. The corresponding dissipation model can be considered as a member of the class of anisotropic diffusion filters. The developed positive diffusion filter, based on an entropy steered coefficient correction, is nearly oscillation-free and comparable to the application of limiter functions. The class of anisotropic diffusion filters is extended to systems of conservation laws, namely the Euler equations. This is done by a characteristic splitting where the filter algorithm is applied to each characteristic field separately. Thus, it is shown that the extension of the class of anisotropic dissipation filter to systems of equations is possible in principle.