Rusanov-type Schemes for Hyperbolic Equations: Wave-Speed Estimates, Monotonicity and Stability

TL;DR

This work investigates how errors in wave-speed estimates affect Rusanov-type fluxes for hyperbolic equations, showing that underestimating speeds can break monotonicity and shrink stability ranges, while overestimating speeds preserves monotonicity and enlarges stability. By expressing Rusanov-type schemes through a single speed parameter $\hat{s}=\beta\lambda$, the authors unify several classical schemes (Godunov, Lax-Wendroff, LF, FORCE, FORCE-$\alpha$) and derive precise monotonicity and stability conditions in 1D and 2D. They provide both theoretical analysis and a numerical example to illustrate the impact of speed perturbations, and demonstrate how 2D extensions modify stability regions. The findings offer practical guidance for wave-speed estimation in HLL-type fluxes and motivate further studies for nonlinear, multi-wave systems such as HLL/HLLC. Overall, overestimating wave speeds emerges as advantageous for robustness and stability, with implications for numerical methods in complex hyperbolic systems.

Abstract

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 of these schemes rely on wave speed estimates. Recent work 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 could 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. Our reported findings may also motivate subsequent studies for complex non-linear hyperbolic systems, requiring estimates for two or more waves, such as in HLL and HLLC schemes.

Figures (9)

• Figure 1: Representation of Rusanov-type schemes in the $x$-$t$ plane. Here FTCS is the Forward in Time Centred in Space scheme; LW is Lax-Wendroff; GodU is Godunov upwind; FORCE is the FORCE scheme and LF is Lax-Friedrichs. The three cases of the Godunov centred scheme are represented by GC1, GC2 and GC3. Each scheme corresponds to a particular choice of the characteristic line $\frac{x}{t}=\hat{s}$ emerging from the origin $0$. All linearly stable schemes lie in the wedge $LW0LF$. Large values of $\hat{s}$ are associated with monotone, more diffusive schemes, while low values of $\hat{s}$ are associated with non-monotone schemes.
• Figure 2: The stability region of the class of Rusanov-type schemes lies between the Lax-Wendroff ($\beta=c$) and the Lax-Friedrichs schemes ($\beta=1/c$), and is defined by $c \le \beta \le 1/c$. The region of monotone schemes (blue) lies between the Godunov upwind ($\beta=1$) and the Lax-Friedrichs ($\beta=1/c$) schemes, that is $1 \le \beta \le 1/c$. The region of non-monotone but stable schemes (salmon) lies between Lax-Wendroff ($\beta=c$) and Godunov upwind ($\beta=1$), that is $c \le \beta < 1$. Note that the region of monotone schemes (blue) extends to $\infty$ as $c$ tends to zero. The rest of the rectangular region (white) identifies unconditionally unstable schemes.
• Figure 3: Beta functions for conventional methods represented as Rusanov-type schemes. The region of stable schemes is bounded below by the Lax-Wendroff line $\beta(c)=c$ and bounded above by the Lax-Friedrichs curve $\beta(c)= 1/c$. As also shown in Figure \ref{['fig:FullStability']} the full region of stable schemes is divided into two sub-regions by the horizontal line $\beta=1$ corresponding to the Godunov upwind scheme. Two more linearly stable schemes are the FORCE scheme given by the curve $\beta(c) = (1+c^{2})/2c$ and the Godunov centred scheme is given by the line $\beta(c)=2c$. The FORCE method is monotone in the full range of Courant numbers. The Godunov centred scheme is non-monotone for $c \le 1/2$ but monotone for $1/2 < c \le \frac{1}{2} \sqrt{2}=c_{lim}$; the upper limit for $c_{lim}$ is given by the intersection of the Godunov centred scheme with the Lax-Friedrichs scheme.
• Figure 4: $\beta(c;\alpha)$ functions for the FORCE-$\alpha$ schemes, along with three conventional monotone schemes, namely Godunov upwind, Lax-Friedrichs and the conventional FORCE method ($\alpha=1$). $\beta(c; \alpha)$ functions for the FORCE-$\alpha$ schemes are shown for $\alpha \in \left\{2,3,4,5 \right\}$. The stability limit $c_{lim}$ is given by the point of intersection between the $\beta(c;\alpha)$ and the $\beta_{LF}=1/c$ curves. As $\alpha$ increases $c_{lim}$ decreases.
• Figure 5: Rusanov-type schemes resulting from constant perturbations of the Godunov upwind scheme ($\beta=1$) of the form $\beta = 1-\epsilon_{B}$ (underestimate) and $\beta = 1+\epsilon_{T}$ (overestimate). Two examples shown correspond to underestimation of $\hat{s}$, namely $\epsilon_{B}=0.5$ ($\beta=0.5$) and $\epsilon_{B}=0.25$ ($\beta=0.75$). Two more examples correspond to overestimation of $\hat{s}$, namely $\epsilon_{T}=0.5$ ($\beta=1.5$) and $\epsilon_{T}=1.0$ ($\beta=2.0$).