Increasing the order of approximation of Godunov's scheme using solutions of the generalized riemann problem

Solvers for the high-order Riemann problem for hyperbolic balance laws

“…For the derivative Riemann problem DRP K , with K > 0, finding the solution in the half plane x 2 ðÀ1; 1Þ; t > 0, is a formidable task that is possible only in special cases. See [28] for the complete solution of the DRP 1 for the Euler equations for ideal gases.…”

“…In the DRP 0 all first and higher-order spatial derivatives of the initial condition away from the origin vanish identically; this case corresponds to the classical piece-wise constant data Riemann problem, associated with the first-order Godunov scheme [15]. Similarly, in the DRP 1 all second and higher-order spatial derivatives of the initial condition for the DRP away from the origin vanish identically; this case corresponds to the piece-wise linear data Riemann problem, or the so-called generalized Riemann problem (GRP), associated with a second-order method of the Godunov type [29,47,1,6,5,28,40,2]. Fig.…”

ADER schemes for three-dimensional non-linear hyperbolic systems

“…The remaining higher order time derivatives of the flux in (23) are expressed via time derivatives of the intercell state Q(x i + 1/2 , y a , z b , s) which are given by the Taylor expansion (18). The solution is advanced in time by updating the cell averages according to the one-step formula (14).…”

“…The corresponding generalised Riemann problem (GRP) schemes in one space dimension have been constructed by various authors, e.g. [2,14,16].…”

“…Instead, it is replaced by two conventional Riemann problems, namely one non-linear problem for the leading term for state variables and one linear problem for gradients of state variables. Since approximate-state Riemann solvers are available for most of the hyperbolic conservation laws of interest, the MGRP scheme is much more practical than the original GRP schemes [2,14,16].…”