The generalized Riemann problems for hyperbolic balance laws: A unified formulation 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. 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 [M. Ben-Artzi and J. Falcovitz, "Generalized Riemann Problems in Computational Fluid Dynamics", Cambridge University Press, 2003]. A more general formulation of the second-order GRP solver is still confined with a class of weakly coupled systems [Numer. Math. (2007) 106:369-425]. This paper provides a unified approach for solving the GRP in the general context of hyperbolic balance laws, without weakly coupled constraint, towards high order accuracy. The derivation of the second-order GRP solver is more concise compared to those in previous works and the third-order 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 the rarefaction wave by deriving the L(Q)-equations, an ODE system capturing the "evolution" of the characteristic derivatives in x-t space for generalized Riemann invariants. The case of a sonic point is incorporated into a general treatment. The accuracy of the derived GRP solvers are justified and numerical examples are presented for the performance of the resulting scheme.