Hyperbolic balance laws: Riemann invariants and the generalized Riemann problem

Abstract

The Generalized Riemann Problem (GRP) for a nonlinear hyperbolic system of m balance laws (or alternatively "quasi-conservative" laws) in one space dimension is now well-known and can be formulated as follows: Given initial-data which are analytic on two sides of a discontinuity, determine the time evolution of the solution at the discontinuity. In particular, the GRP numerical scheme (second-order high resolution) is based on an analytical evaluation of the first time derivative. It turns out that this derivative depends only on the first-order spatial derivatives, hence the initial data can be taken as piecewise linear. The analytical solution is readily obtained for a single equation (m = 1) and, more generally, if the system is endowed with a complete (coordinate) set of Riemann invariants. In this case it can be "diagonalized" and reduced to the scalar case. However, most systems with m > 2 do not admit such a set of Riemann invariants. This paper introduces a generalization of this concept: weakly coupled systems (WCS). Such systems have only "partial set" of Riemann invariants, but these sets are weakly coupled in a way which enables a "diagonalized" treatment of the GRP. An important example of a WCS is the Euler system of compressible, nonisentropic fluid flow (m = 3). The solution of the GRP discussed here is based on a careful analysis of rarefaction waves. A "propagation of singularities" argument is applied to appropriate Riemann invariants across the rarefaction fan. It serves to "rotate" initial spatial slopes into "time derivative". In particular, the case of a "sonic point" is incorporated easily into the general treatment. A GRP scheme based on this solution is derived, and several numerical examples are presented. Special attention is given to the "acoustic approximation" of the analytical solution. It can be viewed as a proper linearization (different from the approach of Roe) of the nonlinear system. The resulting numerical scheme is the simplest (second-order, high-resolution) generalization of the Godunov scheme.

Implementation of the GRP scheme for computing radially symmetric compressible fluid flows

“…Thus the present scheme is directly Eulerian with easy extension to multi-dimensions and no special treatment is required for sonic cases. Indeed, this technique was already used for shallow water equations [20], compressible fluid flows [8] and more general systems [7]. The further advantage of this approach is that the source term (geometrical) effect is included in numerical fluxes.…”

Comparison of the generalized Riemann solver and the gas-kinetic scheme for inviscid compressible flow simulations

“…Due to the regularity of ψ and S and their analytical resolution, it is expected that the rarefaction wave can be captured very accurately using the GRP scheme, and it is indeed. The detail can be found in [6,5].…”

“…In order to further increase the accuracy, generalized Riemann solvers under piecewise linear discontinuous initial data were developed. The generalized Riemann problem (GRP) was proposed for compressible flows based on the Lagrangian formulation first [2,4], and a direct Eulerian version was developed in [6,5] using the concept of Riemann invariants. Theoretically, in the GRP, a close coupling between the spatial and temporal evolution is recovered through the analysis of detailed wave interactions.…”

Remark on the generalized Riemann problem method for compressible fluid flows

“…This concept was used in order to derive a direct Eulerian scheme [9,6], which avoids the passage from the Lagrangian to the Eulerian framework. This concept can also be extended to a more general setting of hyperbolic balance laws [5].…”