Oscillation-eliminating central DG schemes for hyperbolic conservation laws

Abstract

This paper proposes and analyzes a class of essentially non-oscillatory central discontinuous Galerkin (CDG) methods for general hyperbolic conservation laws. First, we introduce a novel compact, non-oscillatory stabilization mechanism that effectively suppresses spurious oscillations while preserving the high-order accuracy of CDG methods. Unlike existing limiter-based approaches that rely on large stencils or problem-specific parameters for oscillation control, our dual damping mechanism is inspired by CDG-based numerical dissipation and leverages overlapping solutions within the CDG framework, significantly enhancing stability while maintaining compactness. Our approach is free of problem-dependent parameters and complex characteristic decomposition, making it efficient and robust. Second, we provide a rigorous stability and optimal error analysis for fully discrete Runge-Kutta (RK) CDG schemes, addressing a gap in the theoretical understanding of these methods. Specifically, we establish the approximate skew-symmetry and weak boundedness of the CDG discretization. These results enable us to rigorously analyze the fully discrete error estimates for our oscillation-eliminating CDG (OECDG) method, a challenging task due to its nonlinear nature, even for linear advection equations. Building on this framework, we reformulate nonlinear oscillation-eliminating CDG schemes as linear RK CDG schemes with a nonlinear source term, extending error estimates beyond the linear case to schemes with nonlinear oscillation control. While existing error analyses for DG or CDG schemes have largely been restricted to linear cases without nonlinear oscillation-control techniques, our analysis represents an important theoretical advancement. Experiments validate the theoretical findings and demonstrate the effectiveness of the OECDG method.