[Paper Review] On the Accuracy and Stability of Various DG Formulations for Diffusion

Mohammad Alhawwary, Z.J. Wang|arXiv (Cornell University)|Oct 7, 2018

Computational Fluid Dynamics and Aerodynamics50 references4 citations

TL;DR

This paper analyzes the accuracy and stability of popular discontinuous Galerkin (DG) formulations for diffusion—SIPG, BR1, BR2, and LDG—using von Neumann and combined-mode analysis. It shows that LDG offers superior dissipation accuracy for high wavenumbers and long-time simulations, while penalty parameter tuning enhances BR1 stability and performance, approaching BR2 behavior.

ABSTRACT

In this paper, we study the stability (in terms of the maximum time step) and accuracy (in terms of the wavenumber-diffusion properties) for several popular discontinuous Galerkin (DG) viscous flux formulations. The considered methods include the symmetric interior penalty formulation (SIPG), the first and second approaches of Bassi and Rebay (BR1, BR2), and the local discontinuous Galerkin method (LDG). For the purpose of stability, we consider the von Neumann stability analysis method for uniform grids with a periodic boundary condition. In addition, the combined-mode analysis approach previously introduced for the wave equation is utilized to analyze the dissipative error. This new approach can be used to study the performance of a particular DG and Runge-Kutta DG (RKDG) scheme for the entire extended wavenumber range. Thus, more insights into the robustness as well as accuracy and efficiency can be obtained. For instance, the LDG method provides larger dissipation for high-wavenumber components than the BR1 and BR2 approaches for short time simulations in addition to a lower error bound for long time simulations. The BR1 approach with added dissipation can have desirable properties and stability similar to BR2. For BR2, the penalty parameter can be adjusted to enhance the performance of the scheme. The results are verified through canonical numerical tests.

Motivation & Objective

• To evaluate the stability and accuracy of major DG formulations (SIPG, BR1, BR2, LDG) for diffusion problems.
• To investigate the impact of the penalty parameter on time-step stability and dissipation error in fully discrete RKDG schemes.
• To apply combined-mode analysis to assess wavenumber-dependent dissipation across the full wavenumber spectrum.
• To provide closed-form expressions for simplified implementation and to clarify connections among schemes.
• To verify results via numerical tests on linear heat and Burgers' equations.

Proposed method

• Conducts von Neumann stability analysis on uniform grids with periodic boundary conditions to determine maximum stable time steps.
• Applies combined-mode analysis to evaluate dissipation error across the entire wavenumber range, enabling insight into long- and short-time behavior.
• Performs semi-discrete and fully-discrete analysis for RKDG schemes, linking penalty parameters to stability and accuracy.
• Derives simplified closed-form expressions for key DG formulations to aid implementation and reveal structural similarities.
• Uses canonical test cases: linear heat equation with Gaussian initial condition and decaying Burgers' turbulence, with exact solutions for validation.
• Compares schemes across polynomial orders (p = 1 to 5) and time integration schemes (RK2, RK3, RK4).

Experimental results

Research questions

• RQ1 How do SIPG, BR1, BR2, and LDG methods compare in terms of maximum stable time step for RKDG schemes?
• RQ2 What is the influence of the penalty parameter on the dissipation error and stability of BR1 and BR2 schemes?
• RQ3 How does the LDG method perform in approximating exact dissipation behavior for high wavenumbers over long-time simulations?
• RQ4 Can a stabilized BR1 formulation achieve dissipation properties similar to BR2 through penalty parameter adjustment?
• RQ5 How do short-time and long-time dissipation behaviors differ across schemes, and what explains the observed discrepancies?

Key findings

• The LDG method provides larger dissipation for high-wavenumber components than BR1 and BR2 in short-time simulations, with a lower error bound in long-time simulations.
• For BR2, adjusting the penalty parameter η significantly improves scheme performance, with optimal values enhancing stability and accuracy.
• The BR1 scheme with added dissipation (η ≈ 1.33 for p=2) achieves dissipation behavior nearly identical to BR2 (η=2) in both semi-discrete and fully-discrete cases.
• At approximately 50% of the maximum stable time step, all schemes recover their semi-discrete dissipation behavior; near Δτ_max, all exhibit slower-than-exact decay for high wavenumbers.
• For the 1D linear heat equation, BR2 schemes have Δτ_max values about 2.5–3.1 times larger than LDG for the same polynomial order and time integration.
• In the decaying Burgers' turbulence test, LDG-η0 best captures the energy spectrum, confirming superior accuracy in practical, nonlinear settings.