High-order accurate solutions of generalized Riemann problems of nonlinear hyperbolic balance laws

We provide a systematic study for the generalized Riemann problem(GRP)of the nonlinear hyperbolic balance law,which is critically concerned with the construction of the spatial-temporally coupled high-order Godunov-type scheme.The full analytical GRP solvers up to the third-order accuracy and also a collection of properties of the GRP solution are derived by resolving the elementary waves.The resolution of the rarefaction wave is a crucial point,which relies on the use of the generalized characteristic coordinate(GCC)to analyze the solution at the singularity.From the analysis on the GCC,we derive for the general nonlinear system the evolutionary equations for the derivatives of generalized Riemann invariants.For the nonsonic case,the full set of spatial and temporal derivatives of the GRP solution at the singularity are obtained,whereas for the sonic case the limiting directional derivatives inside the rarefaction wave are derived.In addition,the acoustic approximation of the analytical GRP solver is deduced by estimating the error it introduces.It is shown that the computationally more efficient Toro-Titarev solver can be the approximation of the analytical solver under the suitable condition.Hence this work also provides a theoretical basis of the approximate GRP solver.The theoretical results are illustrated via the examples of the Burgers equation,the shallow water equations and a system for compressible flows under gravity acceleration.Numerical results demonstrate the accuracy of the GRP solvers for both weak and strong discontinuity cases.