Computing Slowly Moving Shocks with Flux-differencing Schemes
The numerical error produced when computing slowly moving shocks in a finite volume framework is studied. The error is a result of discretizing a thin shock profile dynamic on the scale of spatial and temporal discretization. It has been studied in great detail by many practitioners, but no fix has been built to totally eradicate the problem unique to nonlinear hyperbolic systems. I argue that current shock capturing schemes do not accurately resolve the nonlinear shock profile and this results in erroneous oscillations. I study numerical tools used to pursue high-order accurate solutions of slow shocks, including a hybridized upwind-biased slope limiter for the piecewise parabolic method, a universal Osher Riemann solver, and a new weighted essentially non-oscillatory (WENO)-type slope limiter. The effectiveness of these tools are tested in two hyperbolic systems: the Euler equations and the ideal magnetohydrodynamics equations. The success of the upwind-biased slope limiter has been verified against high-order accurate schemes such as WENO.