OEDG: Oscillation-Eliminating Discontinuous Galerkin Method for Hyperbolic Conservation Laws

Abstract

Controlling spurious oscillations is crucial for designing reliable numerical schemes for hyperbolic conservation laws. In this talk, we propose a novel, robust, and efficient oscillation-eliminating discontinuous Galerkin (OEDG) method on general meshes, motivated by the damping technique in [Lu, Liu, and Shu, SIAM J. Numer. Anal., 59:1299–1324, 2021]. The OEDG method incorporates an OE procedure after each Runge–Kutta stage, and it is devised by alternately evolving the conventional semidiscrete DG scheme and a damping equation. A novel damping operator is designed to possess both scale-invariant and evolution-invariant properties. We rigorously prove optimal error estimates of the fully discrete OEDG method for smooth solutions of linear scalar conservation laws. The OEDG method exhibits many notable advantages. It effectively eliminates spurious oscillations for problems spanning various scales and wave speeds without problem-specific parameters. Furthermore, it retains the key properties of the conventional DG method, such as conservation, optimal convergence rates, and superconvergence. Moreover, the OEDG method maintains stability under the normal CFL condition, even in the presence of strong shocks associated with highly stiff damping terms. The OE procedure is non-intrusive, facilitating seamless integration into existing DG codes as an independent module. Extensive numerical results confirm the analysis and validate the effectiveness of the OEDG method.