Uniformly Effective Numerical Methods for Hyperbolic Systems

A unified method to compute compressible and incompressible flows is presented. Accuracy and efficiency do not degrade as the Mach number tends to zero. A staggered scheme solved with a pressure correction method is used. The equation of state is arbitrary. A Riemann problem for the barotropic Euler equations with nonconvex equation of state is solved exactly and numericaly. A hydrodynamic flow with cavitation in which the Mach number varies between 10−3 and 20 is computed. Unified methods for compressible and incompressible flows are further discussed for the flow of a perfect gas. The staggered scheme with pressure correction is found to have Mach-uniform accuracy and efficiency, and for the fully compressible case the accuracy is comparable with that of established schemes for compressible flows.