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

TL;DR

A novel, robust, and efficient oscillation-eliminating discontinuous Galerkin (OEDG) method on general meshes, motivated by the damping technique, which reveals the role of damping operator as a modal filter and establishes close relations between the damped and spectral viscosity techniques.

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.

Figures (14)

• Figure 1: Comparisons of third-order OEDG and OFDG methods for problems of different scales and wave speeds. The cell averages of the DG solutions are plotted: $u_h^\lambda$ denotes numerical solutions at $t=1.1$ for $u_t + u_x =0$ with scaled initial data $u(x,0)=\lambda u_0(x)$; $u_h^\beta$ denotes numerical solutions at $t=1.1/\beta$ for $u_t + \beta u_x =0$ with $u(x,0)= u_0(x)$, where $u_0(x)$ is defined in \ref{['eq:u0_exp2']}. See \ref{['ex2']} for the detailed setup. The OFDG method exhibits excessive smearing or persistent spurious oscillations (overshoots or undershoots) in large or small scale cases, while OEDG method performs consistently well thanks to scale invariance and evolution invariance. The results of OFDG method can be improved if our new scale-invariant and evolution-invariant damping is used instead.
• Figure 1: Density at one hour computed by OEDG and OFDG methods solving three equivalent equations in different units for \ref{['ex:tf']}. Cell averages are plotted.
• Figure 2: Density of Lax problem at $t = 1.3$ computed by OEDG and OFDG methods.
• Figure 3: Density of Woodward--Colella blast wave problem at $t=0.038$ computed by OEDG and OFDG methods with or without characteristic decomposition. Solution polynomials are plotted.
• Figure 4: Density of Shu--Osher problem at $t = 1.8$ computed by OEDG and OFDG methods. Solution polynomials are plotted.

Theorems & Definitions (50)

• Remark 2.1: dimensionless damping
• Remark 2.2: conservation
• Remark 2.3: stability
• Remark 2.4: simplicity and efficiency
• Remark 2.5
• Theorem 2.6: scale invariance
• Theorem 2.7: homogeneity
• Proof 1
• Remark 2.8
• Remark 2.9: local scale invariance