A contribution to the great Riemann solver debate

Abstract

The aims of this paper are threefold: to increase the level of awareness within the shock‐capturing community of the fact that many Godunov‐type methods contain subtle flaws that can cause spurious solutions to be computed; to identify one mechanism that might thwart attempts to produce very‐high‐resolution simulations; and to proffer a simple strategy for overcoming the specific failings of individual Riemann solvers.

A Nonlinear Flux Split Method for Hyperbolic Conservation Laws

“…The method is combined with piecewise hyperbolic reconstruction (which uses hyperbola as reconstructing functions, see [14]) in space, and the total variation diminishing Runge-Kutta (RK) methods in time [20], to construct a high-order version of the firstorder scheme. This new approach preserves positivity, provides accurate, oscillation-free, and well-behaved numerical approximation, and fixes a variety of numerical pathologies mentioned in [16].…”

“…The initial data we consider correspond to a Mach-3 shock moving to the right with a speed s = 0.1096 [16],…”

“…In [1,17,16] and references therein, it has been observed that the numerical computation of slowly moving shock waves displays an anomalous behavior inherent to nonlinear systems: the numerical values in the postshock region display an oscillatory behavior which is completely nonphysical. Numerical results computed by the present, Marquina, and HLLE schemes are displayed in Fig.…”

“…With an appropriate choice of particle velocities, Einfeldt [4] has shown that this scheme (denoted HLLE) satisfies all the important properties met by the Godunov method (e.g., positivity and entropy conditions). However it lacks information about isolated contact discontinuities [16].…”

“…A different strategy [16] is to combine some desirable and complementary features of two or more solvers (e.g., a FDS and FVS). With this approach it is possible to construct a less-dissipative upwind scheme, and to control certain instabilities by changing the flavor of the dissipation mechanism rather than by using judiciously a small amount of artificial dissipation.…”