On the Strong Stability Preserving Property of Runge-Kutta Methods for Hyperbolic Problems

TL;DR

The paper addresses stability of Runge-Kutta time integration for hyperbolic problems and expands beyond the SSP framework by introducing a modified RK representation that ties stability of the forward Euler step to general RK schemes with coefficients in $[0,1]$. Under this framework, the authors prove that non-SSP RK methods can preserve entropy stability for strictly convex functionals, as well as positivity of density and internal energy and TVD properties for Burgers' equation with Lax-Friedrichs, first-order upwind, and MUSCL discretizations, provided the time step is sufficiently small relative to a forward-Euler limit $\Delta t_{FE}$. Numerical experiments across energy-dissipative Burgers, upwind, MUSCL, and Leblanc shock-tube tests corroborate the theory, showing nonzero stability-time-step limits $c^p$ and $c^s$ even for non-SSP schemes like RK44. These results broaden the practical toolkit for high-order time integration in hyperbolic PDEs, suggesting that safer or more efficient RK choices may be feasible beyond the traditional SSP subset. The work highlights a path toward estimating problem-specific stability limits and extending the analysis to additional spatial discretizations.

Abstract

Strong Stability Preserving (SSP) time integration schemes maintain stability of the forward Euler method for any initial value problem. However, only a small subset of Runge-Kutta (RK) methods are SSP, and many efficient high-order time integration schemes do not formally belong to this class. In this work, we introduce a mathematical strategy to analyze the nonlinear stability of RK schemes that may not necessarily belong to the SSP class. With this approach, we mathematically demonstrate that there are time integration schemes outside the class of SSP schemes that can maintain entropy stability and positivity of density and pressure for the Lax-Friedrichs discretization, and Total Variation Diminishing stability for the first-order upwind and the second-order MUSCL schemes. As a result, for these problems, a broader range of RK methods, including the classical fourth-order, four-stage RK scheme, can be used while the numerical integration remains stable. Numerical experiments confirm these theoretical findings, and additional experiments demonstrate similar observations for a wider class of space discretizatins.

Figures (9)

• Figure 1: Energy dissipative Burgers' problem integrated with RK44 using $\Delta t= \Delta t_{FE}$. This figure shows that both RK solution and $\underline{q}^n + \Delta t \underline{R}^j$ term preserved energy dissipative behavior of forward Euler method with a nonzero step size.
• Figure 2: Burgers' problem with a first-order upwind spatial discretization, integrated with RK44 using $\Delta t= \Delta t_{FE}$. It demonstrates that the TVD property is maintained with this non-SSP method using a nonzero time-step.
• Figure 3: $TV\left(\underline{q}^n + \Delta t \underline{R}^{j=s} \right) - TV \left(\underline{q}^n \right)$ values of the RK44 scheme with $\Delta t= \Delta t_{FE}$, for the Burgers' problem with a first-order upwind spatial discretization. It shows that $\underline{q}^n + \Delta t \underline{R}^{j=s}$ terms maintain the TVD property of forward Euler method with a nonzero time-step .
• Figure 4: Burgers' problem discretized with the second-order MUSCL scheme, integrated with RK44 using $\Delta t= \Delta t_{FE}$, showing that RK44 maintains the TVD property of the forward Euler solution.
• Figure 5: $TV\left(\underline{q}^n + \Delta t \underline{R}^{j=s} \right) - TV \left(\underline{q}^n \right)$, when the Burgers' problem is discretized with the second-order MUSCL scheme and integrated with RK44 using $\Delta t= \Delta t_{FE}$.

Theorems & Definitions (26)

• Remark 1
• Lemma 1
• proof : Proof of Lemma \ref{['lemma_SSP_modified_RK']}
• Lemma 2
• proof : Proof of Lemma \ref{['lemma_convex_functional']}
• Theorem 3
• proof : Proof of Theorem \ref{['Theorem_entropy_stability']}
• Lemma 4
• proof : Proof of Lemma \ref{['lemma_positivity_density']}
• Theorem 5