Chebyshev pseudospectral methods for conservation laws with source terms and applications to multiphase flow

Pseudospectral methods are well known to produce superior results for the solution of partial differential equations whose solutions have a certain amount of regularity. Recent advances have made possible the use of spectral methods for the solution of conservation laws whose solutions may contain shocks. We use a recently described Super Spectral Viscosity method to obtain stable approximations of Systems of Nonlinear Hyperbolic Conservation Laws. A recently developed postprocessing method, which is theoretically capable of completely removing the Gibbs phenomenon from the Super Spectral Viscosity approximation, is examined. The postprocessing method has shown great promise when applied in some simple cases. We discuss its application to more complicated problems and examine the possibility of the method being used as a “black box” postprocessing method. Applications to multiphase fluid flow are made.