Rusanov-type schemes for hyperbolic equations: Wave-speed estimates, monotonicity and stability

HLL-type schemes constitute a large hierarchy of numerical methods, in the finite volume and discontinuous Galerkin finite element frameworks, for solving hyperbolic equations. The hierarchy of fluxes includes Rusanov schemes, HLL schemes, HLLC schemes, and other variations. All these schemes rely on wave speed estimates. Recent work [1] has shown that most wave speed estimates in current use underestimate the true wave speeds. In the present paper we carry out a theoretical study of the consequences arising from errors in the wave speed estimates, on the monotonicity and stability properties of the derived schemes. For the simplest case of the hierarchy, that is Rusanov-type schemes, we carry out a detailed analysis in terms of the linear advection equation in one and two space dimensions. It is found that errors from underestimates of the wave speed cause loss of monotonicity, severe reduction of the stability limit, and even loss of stability. Errors from overestimates, though preserving monotonicity, will cause a reduction of the stability limit. We find that overestimation is preferable to underestimation, for two reasons. First, schemes from overestimation are monotone, and second, their stability regions are larger than those from underestimation, for equivalent displacements from the exact speed. The findings of this paper may prove useful in raising awareness of the potential pitfalls of a seemingly simple practical computational task, that of providing wave speed estimates. Finally, we find that, on the whole, some aspects of the linear theory extend to non-linear equations. Our reported findings may motivate subsequent studies for complex non-linear hyperbolic systems, requiring estimates for one, two or more waves, such as in Rusanov, HLL and HLLC schemes, as well as for reliable computation of the time step in explicit update formulae.