Sub-grid properties and artificial viscous stresses in staggered-mesh schemes

Abstract

We analyze the properties of a number of forms of artificial viscosity, when applied to the Lagrangian phase of a staggered grid Lagrange–remap hydrodynamics code, using a full von Neumann stability analysis. This allows us to derive the numerical dispersion relation for a single step of the scheme.We study the development of shocks within a weakly nonlinear acoustic wave in detail. The second-order von Neumann–Richtmyer artificial viscosity leads to significant post-shock ringing, while adding linear artificial viscosity reduces the accuracy of the scheme to first order and introduces a dependence on flow Mach number. The form of the numerical dispersion relation suggests that to control the post-shock ringing without reducing the accuracy of the solution, a higher-order artificial viscosity is required which will preferentially damp modes close to the mesh scale which are subject to significant phase error.The application of ideas based on the Large Eddy Simulation technique for the modelling of turbulent flows leads to a form of artificial viscous stress similar to that described by Schulz (1964). This artificial stress has several features which are beneficial for accurate modelling of compressible turbulent mixing processes, as demonstrated by a range of other test problems.We show that, in the limit of smooth flows, Christensen's approach of applying limiters to the components of a low-order artificial viscosity is analytically equivalent to a high-order scheme. However, the directionally-split implementation suggested in the original publication leads to damping of low Mach number Kelvin–Helmholtz roll up problems similar to that previously observed for finite-volume Godunov schemes.