A high resolution Lagrangian method using nonlinear hybridization and hyperviscosity

Abstract

Classical artificial viscosity methods often suffer from excessive numerical viscosity both at and away from shocks. While a proper amount of dissipation is necessary at the shock wave, it should be minimized away from the shock and disappear where the flow is smooth. The common approach to remove the excessive dissipation is to introduce a limiter. We use a limiting methodology based on nonlinear hybridization, which generalizes to multiple dimensions naturally. Moreover, the properties of the limiter are made mesh independent through abiding by important symmetry and invariance characteristics. A secondary impact of the approach is the use of more optimal coefficients for the viscosity itself. The coefficients can be derived directly through analysis of the Rankine–Hugoniot relations. We can further refine our approach with the use of hyperviscous dissipation that helps to more effectively control oscillations. The hyperviscosity is defined by applying a filter to the original unlimited viscosity, which is then combined using the original limiter. The combination of the limiter with the hyperviscosity produces sharp shock transitions while effectively reducing the amount of high frequency noise emitted by the shock. These characteristics are demonstrated computationally and we show that the limiter returns the overall method to second-order accuracy with or without the contribution of the hyperviscosity.