Application of the Chebyshev pseudospectral method to van der Waals fluids

In this paper, we consider a class of van der Waals flows with non-convex flux functions. In these flows, nonclassical under-compressive shock waves can develop. Such waves, which are characterized by kinetic functions, violate classical entropy conditions. We propose to use a Chebyshev pseudospectral method for solving the governing equations. A comparison of the results of this method with very high-order (up to 10th-order accurate) finite difference schemes is presented, which shows that the proposed method leads to a lower level of numerical oscillations than other high-order finite difference schemes and also does not exhibit fast-traveling packages of short waves which are usually observed in high-order finite difference methods. The proposed method can thus successfully capture various complex regimes of waves and phase transitions in both elliptic and hyperbolic regimes.