An Efficient, Second Order Accurate, Universal Generalized Riemann Problem Solver Based on the HLLI Riemann Solver

摘要

The Riemann problem, and the associated generalized Riemann problem, areincreasingly seen as the important building blocks for modern higher orderGodunov-type schemes. In the past, building a generalized Riemann problemsolver was seen as an intricately mathematical task for complicated physical orengineering problems because the associated Riemann problem is different foreach hyperbolic system of interest. This paper changes that situation. The HLLI Riemann solver is a recently-proposed Riemann solver that isuniversal in that it is applicable to any hyperbolic system, whether inconservation form or with non-conservative products. The HLLI Riemann solver isalso complete in the sense that if it is given a complete set of eigenvectors,it represents all waves with minimal dissipation. It is, therefore, veryattractive to build a generalized Riemann problem solver version of the HLLIRiemann solver. This is the task that is accomplished in the present paper. Weshow that at second order, the generalized Riemann problem version of the HLLIRiemann solver is easy to design. Our GRP solver is also complete and universalbecause it inherits those good properties from original HLLI Riemann solver. Wealso show how our GRP solver can be adapted to the solution of hyperbolicsystems with stiff source terms. Our generalized HLLI Riemann solver is easy to implement and performsrobustly and well over a range of test problems. All implementation-relateddetails are presented. Results from several stringent test problems are shown.These test problems are drawn from many different hyperbolic systems, andinclude hyperbolic systems in conservation form; with non-conservativeproducts; and with stiff source terms. The present generalized Riemann problemsolver performs well on all of them.