OEDG: Oscillation-eliminating discontinuous Galerkin method for hyperbolic conservation laws

TL;DR

This paper introduces a new oscillation-eliminating discontinuous Galerkin method (OEDG) for hyperbolic conservation laws that effectively suppresses spurious oscillations, maintains key properties, and is easy to implement on general meshes.

Contribution

The paper presents a novel OEDG method with a scale-invariant damping operator, providing the first generic fully-discrete error estimates for nonlinear DG schemes with automatic oscillation control.

Findings

Effectively eliminates spurious oscillations across various problems.

Retains conservation, optimal convergence, and superconvergence properties.

Demonstrates stability under standard CFL conditions.

Abstract

Controlling spurious oscillations is crucial for designing reliable numerical schemes for hyperbolic conservation laws. This paper proposes 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, devised by alternately evolving conventional semidiscrete DG scheme and a damping equation. A novel damping operator is carefully designed to possess scale-invariant and evolution-invariant properties. We rigorously prove optimal error estimates of the fully discrete OEDG method for linear scalar conservation laws. This might be the first generic fully-discrete error estimates for nonlinear DG schemes with automatic oscillation control mechanism. The OEDG method exhibits many notable advantages. It effectively eliminates spurious oscillations for challenging problems across various scales and wave speeds, without problem-specific parameters. It obviates the need for characteristic decomposition in hyperbolic systems. It retains key properties of conventional DG method, such as conservation, optimal convergence rates, and superconvergence. Moreover, it remains stable under normal CFL condition. The OE procedure is non-intrusive, facilitating integration into existing DG codes as an independent module. Its implementation is easy and efficient, involving only simple multiplications of modal coefficients by scalars. The OEDG approach provides new insights into the damping mechanism for oscillation control. It reveals the role of damping operator as a modal filter and establishes close relations between the damping and spectral viscosity techniques. Extensive numerical results confirm the theoretical analysis and validate the effectiveness and advantages of the OEDG method.