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

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.