High-Order Schemes for Multifluid Problems

Geoffrey Main, Stanford University

Efficient numerical schemes for simulating multi-phase hydrodynamic flow under extreme conditions are important for stewardship science. Although we have made strides, our methods as originally designed for these problems are, as nearly all existing Eulerian multifluid methods are, only formally first-order accurate at the interface. This typically degrades the global order of convergence to only 1.4-1.5, even if the scheme is second-order everywhere else. Unfortunately, this means that for these problems, especially ones involving extreme conditions, a finer mesh is required to obtain the same level of accuracy than if the scheme were second-order accurate everywhere.

It is possible to improve the accuracy by constructing schemes for these problems that are formally high-order accurate even at the interface. We have worked on modifying existing Eulerian schemes for multifluid problems, which are only first-order accurate at material interfaces, in order to achieve higher-order accuracy at these boundaries. We describe our scheme in one dimension for problems in both Cartesian and spherical coordinates and then elucidate an extension to three dimensions. Our numerical results demonstrate empirically that these methods are indeed second-order accurate for both explicit and implicit time stepping. Although our current results are for a second-order accurate treatment of the interface, the interpolation/extrapolation method is very general and can achieve an arbitrary order of accuracy at the interface, including sixth order.

Abstract Author(s): Alex Main