Fourier analysis and evaluation of DG, FD and compact difference methods for conservation laws

Abstract

Large eddy simulation (LES) has been increasingly used to tackle vortex-dominated turbulent flows. In LES, the quality of the simulation results hinges upon the quality of the numerical discretizations in both space and time. It is in this context we perform a Fourier analysis of several popular methods in LES including the discontinuous Galerkin (DG), finite difference (FD), and compact difference (CD) methods. We begin by reviewing the semi-discrete versions of all methods under-consideration, followed by a fully-discrete analysis with explicit Runge-Kutta (RK) time integration schemes. In this regard, we are able to unravel the true dispersion/dissipation behavior of DG and Runge-Kutta DG (RKDG) schemes for the entire wavenumber range. The physical-mode is verified to be a good approximation for the asymptotic behavior of these DG schemes in the low wavenumber range. After that, we proceed to compare the DG, FD, and CD methods in dispersion and dissipation properties. Numerical tests are conducted using the linear advection equation to verify the analysis. In comparing different methods, it is found that the overall numerical dissipation strongly depends on the time step. Compact difference (CD) and central finite difference (FD) schemes, in some particular settings, can have more numerical dissipation than the DG scheme with an upwind flux. This claim is then verified through a numerical test using the Burgers' equation. 30 22, 24] defined as the one that approximates the exact dispersion relation for a range of wavenumbers while regarding other modes as spurious. Recently, Moura et al. [27] provided new interpretations on the role of spurious or secondary modes. In their work, these modes are replicates of the physical-mode along the wavenumber axis and they improve the accuracy of the scheme. Vanharen et al. [29] concluded that after a large number of iterations, high-order schemes behave in dispersion and dissipation according to the 35 physical-mode asymptotically, for wavenumbers less than π. Nevertheless, the complete behavior of DG-type high-order schemes in dispersion and dissipation based on all eigenmodes has not been studied before. In this paper, we provide a first attempt to achieve this goal.Whilst there exists abundant work on the analysis of both high-order and low-order schemes or classical finite difference/finite-volume schemes, little attention was given to comparing the DG, FD, and CD schemes 40 of the same order of accuracy. The DG method, originally introduced by Reed and Hill [30] to solve the neutron transport equation, is chosen in this study as a representative of the high-order polynomialbased methods capable of handling unstructured grids including the spectral difference (SD) [31], and the flux reconstruction (FR) or correction procedure via reconstruction (CPR) methods [32]. LaSaint and Raviart [33] performed an error analysis for the DG method. It was then further developed for convection-45 dominated problems and fluid dynamics by many researches, see for example ([34, 35, 36,...

A Two-Dimensional Central Non-Oscillatory Scheme for Inviscid Compressible Flows

“…In order to evaluate the summation in Eq. 1, the CENO scheme reconstructs the fluxes at each Gauss quadrature points of the cells' interfaces [9]. To reduce error propagation between each time step, we use a Runge-Kutta 4 th Kutta 4 th order scheme [10].…”

“…High order method in space and time have demonstrated the capabilities of resolving such flows with desired accuracy, by using coarser meshes [6]. In the literature, there are several schemes available, such as essentially non-oscillatory (ENO) schemes [7], Weighted ENO (WENO) schemes [7], Central ENO (CENO) schemes [8], discontinuous Galerkin (DG) schemes [9], and finite-difference (FD) schemes [9]. The main challenge of a high order scheme is the preservation of monotonicity during reconstruction stage.…”

“…In order to choose the most appropriate stencil for reconstruction, the WENO scheme applies a weighting factor to each stencil, which is depending upon the solution [7]. At last, the CENO scheme is using a central stencil to reconstruct the solution, but the order of accuracy is reduced to limited second order where the cells present a discontinuity of any kind [9]. To understand whether a cell is under resolved, we use a smoothness indicator [9].…”

“…At last, the CENO scheme is using a central stencil to reconstruct the solution, but the order of accuracy is reduced to limited second order where the cells present a discontinuity of any kind [9]. To understand whether a cell is under resolved, we use a smoothness indicator [9]. Due to the high possibility of ENO and WENO schemes at selecting the proper stencil, their usage requires an extreme computational cost.…”

“…However, the lower computational load comes along with loss of uniform high accuracy within the whole domain. Researchers, such as Ivan et al [9], proved that this local loss of accuracy is not nullifying the strength of high order accuracy in the whole domain; rather the convergence rate is still at the high order.…”