On Godunov-type methods for gas dynamics
SIAM Journal on Numerical Analysis
Efficient implementation of essentially non-oscillatory shock-capturing schemes
Journal of Computational Physics
SIAM Journal on Mathematical Analysis
On Godunov-type methods near low densities
Journal of Computational Physics
Numerical solution of the Riemann problem for two-dimensional gas dynamics
SIAM Journal on Scientific Computing
Weighted essentially non-oscillatory schemes
Journal of Computational Physics
Efficient implementation of weighted ENO schemes
Journal of Computational Physics
On the Choice of Wavespeeds for the HLLC Riemann Solver
SIAM Journal on Scientific Computing
The Runge-Kutta discontinuous Galerkin method for conservation laws V multidimensional systems
Journal of Computational Physics
Journal of Computational Physics
Compact Central WENO Schemes for Multidimensional Conservation Laws
SIAM Journal on Scientific Computing
Journal of Computational Physics
Resolution of high order WENO schemes for complicated flow structures
Journal of Computational Physics
Essentially Non-Oscillatory and Weighted Essentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws
A multi-state HLL approximate Riemann solver for ideal magnetohydrodynamics
Journal of Computational Physics
Robustness of MUSCL schemes for 2D unstructured meshes
Journal of Computational Physics
Runge-Kutta discontinuous Galerkin method using WENO limiters II: Unstructured meshes
Journal of Computational Physics
On maximum-principle-satisfying high order schemes for scalar conservation laws
Journal of Computational Physics
Journal of Computational Physics
A high-order multi-dimensional HLL-Riemann solver for non-linear Euler equations
Journal of Computational Physics
Self-adjusting, positivity preserving high order schemes for hydrodynamics and magnetohydrodynamics
Journal of Computational Physics
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We present a new HLL-type approximate Riemann solver that aims at capturing any isolated discontinuity without necessitating extensive characteristic analysis of governing partial differential equations. This property is especially attractive for complex hyperbolic systems with more than two equations. Following Linde's (2002) approach [6], we introduce a generic middle wave into the classical two-state HLL solver. The property of this third wave is typified by the way of a ''strength indicator'' that is derived from polynomial considerations. The polynomial that constitutes the basis of the procedure is made non-oscillatory by an adapted fourth-order WENO algorithm (CWENO4). This algorithm makes it possible to derive an expression for the strength indicator. According to the size of this latter parameter, the resulting solver (HLL-RH), either computes the multi-dimensional Rankine-Hugoniot equations if an isolated discontinuity appears in the Riemann fan, or asymptotically tends towards the two-state HLL solver if the solution is locally smooth. The asymptotic version of the HLL-RH solver is demonstrated to be positively conservative and entropy satisfying in its first-order multi-dimensional form provided that a relevant and not too restrictive CFL condition is considered; specific limitations of the conservative increments of the numerical solution and a suited entropy condition enable to maintain these properties in its high-order version. With a monotonicity-preserving algorithm for the time integration, the numerical method so generated, is third order in time and fourth-order accurate in space for the smooth part of the solution; moreover, the scheme is stable and accurate when capturing a shock wave, whatever the complexity of the underlying differential system. Extensive numerical tests for the one- and two-dimensional Euler equation of gas dynamics and comparisons with classical Godunov-type methods help to point out the potentialities and insufficiencies of the method.