A COMBINED-MODE FOURIER ANALYSIS OF DG METHODS FOR LINEAR PARABOLIC PROBLEMS.

Fourier analysis has been shown to provide valuable insight into the dispersion and dissipation characteristics of numerical schemes for PDEs. Applying Fourier analysis to discontinuous Galerkin (DG) methods results in an eigenvalue problem with multiple eigenmodes. It was often relied on one of these modes, the so-called physical-mode, for studying the dispersion and dissipation behavior. The effect of the other modes was considered spurious and typically neglected. Recently, a new approach, the combined-mode approach, was proposed for the linear wave equation, in which all modes are considered. In this paper, we apply the combined-mode approach to a number of DG methods for diffusion. We show that for the linear parabolic heat equation, the physical-mode behavior is completely different than the exact diffusion, over a wide range of wavenumbers, and sometimes nondissipative. In contrast, the combined-mode behavior is more consistent and sufficiently accurate in comparison with the exact diffusion for the whole wavenumber range. This approach also revealed that short time and long time diffusion behaviors are very different for high-order multi--degrees of freedom methods. Additionally, using this approach we conduct a comparative diffusion analysis between a number of popular DG methods for diffusion in one and two dimensions. We also provide a study on the influence of the penalty parameter on their behavior. The considered methods include the symmetric interior penalty, Bassi and Rebay, and the local and compact DG methods. The results are verified numerically through several test cases. [ABSTRACT FROM AUTHOR]