  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.

Numerical Analysis;Computational Physics;Fluid Dynamics