The generalized Riemann problems for compressible fluid flows: Towards high order

The generalized Riemann problems (GRP) for nonlinear hyperbolic systems of balance laws in one space dimension are now well-known and can be formulated as follows: Given initial data which are smooth on two sides of a discontinuity, determine the time evolution of the solution near the discontinuity. In particular, the GRP of (k+1)th order high-resolution is based on an analytical evaluation of time derivatives up to kth order, which turns out to be dependent only on the spatial derivatives up to kth order. While the classical Riemann problem serves as a primary ''building block'' in the construction of many numerical schemes (most notably the Godunov scheme), the analytic study of GRP will lead to an array of ''GRP schemes'', which extend the Godunov scheme. Currently there are extensive studies on the second-order GRP scheme, which proves to be robust and is capable of resolving complex multidimensional fluid dynamic problems (Ben-Artzi and Falcovitz, 2003 [4]). In this paper, we provide a new approach for solving the GRP for the compressible flow system towards high order accuracy. The derivation of second-order GRP solver is more concise compared to those in previous works and the third-order GRP (or quadratic GRP) is resolved for the first time. The latter is shown to be necessary through numerical experiments with strong discontinuities. Our method relies heavily on the new treatment of rarefaction waves. Indeed, as a main technical step, the ''propagation of singularities'' argument for the rarefaction fan, is simplified by deriving the linear ODE systems for the ''evolution'' of ''characteristic derivatives'', in the x-t coordinates, for generalized Riemann invariants. The case of sonic point is incorporated into a general treatment. The accuracy of the derived GRP solvers is justified and numerical examples are presented for the performance of the resulting schemes.