A bound-preserving Runge--Kutta discontinuous Galerkin method with compact stencils for hyperbolic conservation laws

TL;DR

The paper develops a bound-preserving framework for the bound-preserving compact-stencil Runge--Kutta discontinuous Galerkin (cRKDG) method applied to hyperbolic conservation laws, notably the compressible Euler equations. It achieves this by expressing each RK stage as a convex combination of forward-Euler steps involving the standard DG operator and a local derivative operator, and by applying a scaling limiter to enforce pointwise bounds without sacrificing compactness. The authors construct second-, third-, and fourth-order bound-preserving cRKDG schemes with explicit Butcher-form coefficients and demonstrate their effectiveness through 1D and 2D tests, including challenging gas-dynamics benchmarks. The results show robust bound preservation, local conservation, and high-order accuracy, even for large time steps, highlighting the method's practical potential for complex hyperbolic systems and multi-dimensional problems.

Abstract

In this paper, we develop bound-preserving techniques for the Runge--Kutta (RK) discontinuous Galerkin (DG) method with compact stencils (cRKDG method) for hyperbolic conservation laws. The cRKDG method was recently introduced in [Q. Chen, Z. Sun, and Y. Xing, SIAM J. Sci. Comput., 46: A1327--A1351, 2024]. It enhances the compactness of the standard RKDG method, resulting in reduced data communication, simplified boundary treatments, and improved suitability for local time marching. This work improves the robustness of the cRKDG method by enforcing desirable physical bounds while preserving its compactness, local conservation, and high-order accuracy. Our method is extended from the seminal work of [X. Zhang and C.-W. Shu, J. Comput. Phys., 229: 3091--3120, 2010]. We prove that the cell average of the cRKDG method at each RK stage preserves the physical bounds by expressing it as a convex combination of three types of forward-Euler solutions. A scaling limiter is then applied after each RK stage to enforce pointwise bounds. Additionally, we explore RK methods with less restrictive time step sizes. Because the cRKDG method does not rely on strong-stability-preserving RK time discretization, it avoids its order barriers, allowing us to construct a four-stage, fourth-order bound-preserving cRKDG method. Numerical tests on challenging benchmarks are provided to demonstrate the performance of the proposed method.

Figures (8)

• Figure 1: Quadratures in a square cell. From left to right: the quadrature points for edge integrals in Proposition \ref{['assp:edge']} ($\{x_{\beta,\ell,K}\}$), for volume integrals, and for the convex decomposition in Proposition \ref{['assp:CAD']} ($\{x_{\alpha,K},x_{\beta,\ell,K}\}$) for the weak bound-preserving property. The red points are constructed by Gauss quadrature and the black points are constructed by tensor product of Gauss quadrature and Gauss--Lobatto quadrature. The black points are used only in defining the bound-preserving limiter, and they are not used in calculating any numerical integration.
• Figure 2: 1D traveling Heaviside function. Snapshots are taken at $T = 0.5$ on $200$ uniform cells. Top: simulations without applying bound-preserving limiter, which produce solutions with out-of-bound cell averages. Bottom: simulations with bound-preserving limiter and solutions stay within the desired bounds. Only cell averages are plotted.
• Figure 3: Lax shock tube. Simulations with only applying positivity-preserving limiter on $200$ uniform cells. Snapshots are taken at $T = 1.3$. Only cell averages are plotted.
• Figure 4: Double rarefaction. Simulations with only applying positivity-preserving limiter on $200$ uniform cells. Snapshots are taken at $T = 0.6$. Only cell averages are plotted.
• Figure 5: 2D traveling Heaviside function. Simulations with only applying bound-preserving limiter. Snapshots are taken at $T = 1$.

Theorems & Definitions (26)

• Proposition 2.1
• Proposition 2.2
• Proposition 2.3: Weak maximum principle with $\mathcal{F}$
• Proposition 2.4
• Remark 1
• Proposition 2.5: Weak bound-preserving property with $\mathcal{F}$
• Proposition 2.6
• Lemma 3.1: Weak maximum principle with $\pm \mathcal{G}$
• proof
• Proposition 3.2: Lax--Friedrichs splitting property