Designing an Efficient Solution Strategy for Fluid Flows: II. Stable High-Order Central Finite Difference Schemes on Composite Adaptive Grids with Sharp Shock Resolution

Abstract

A simple and efficient solution strategy is designed for fluid flows governed by the compressible Euler equations. It is constructed from a stable high-order central finite difference scheme on structured composite adaptive grids. This basic framework is suitable for solving smooth flows on complicated domains and is easily extendible with extra tools to handle specific flow problems. The stable high-order central difference scheme is mathematically formulated using a recently derived semi-discrete energy method for initial-boundary value problems. The high order of accuracy reduces the number of grid points required in smooth parts of the flow which leads to efficiency in both computational time and memory. A local grid adaptation technique is used to increase the grid density where required. Extra tools are developed for the sharp resolution of shocks. The grids are refined in the shock regions to retain accuracy. On the fine grids in these regions, an effective scalar artificial viscosity term is added to suppress spurious oscillations generated by the high-order central difference method. The location and orientation of shocks is determined by an easy-to-implement wavelet-based detection algorithm. The overhead of the composite adaptive grid method and detection algorithm is negligible compared to the computational kernel. The local grid adaptation with the high-order scheme is shown to increase computational efficiency. The resolution of shocks is sharp.