On explicit discontinuous Galerkin methods for conservation laws

Highlights

• The two explicit DG methods here, the STE-DG and the moment schemes, are based on a ‘predictor-corrector’ formulation.

• The predictor step of the two methods is essentially identical using a Cauchy-Kovalevsky procedure.

• The corrector step also shares the same space-time integration formulation.

• The two methods differ in the evaluation of the projections in the space-time volume integrals.

• The moment scheme has the significant advantage of a CFL condition of 1 in 1D for all degree p and accuracy order of $2p+1$.

• The STE-DG scheme has a CFL condition significantly less than 1 and accuracy order of only $p+1$.

• These schemes, due to the formulation involving no methods of characteristics, extend easily to systems of equations.

• In multiple spatial dimensions, the CFL condition for the moment scheme also becomes restrictive and needs improvement.

Abstract

The concept of projection applied to explicit discontinuous Galerkin (DG) schemes is investigated. The two explicit DG methods in this study are based on a ‘predictor-corrector’ formulation, the first introduced by Lörcher, Gassner, and Munz called space–time expansion discontinuous Galerkin or STE-DG scheme (Lörcher et al. 2007, Gassner et al. 2008), and the second, introduced independently by the author (Huynh 2006, 2013) called the upwind moment scheme. The predictor step of the two methods is essentially identical using a Cauchy-Kovalevsky procedure, which involves no interaction of the data among neighboring cells. The corrector step also shares the same space-time integration formulation and is where interaction takes place; the two methods differ, however, in the evaluation of the projections in the space-time volume integrals. The STE-DG scheme evaluates these in a straightforward manner, whereas the moment scheme employs a successive procedure with each moment update uses the results by the lower-order updates. The trade-off is that the moment scheme has the disadvantage of a more elaborate corrector step and the significant advantage of a CFL (Courant-Friedrichs-Lewy) condition of 1 for all (degree) $p$ and accuracy order of $2p+1$ (super accuracy property) for one-dimensional (1D) advection. In contrast, the STE-DG method is accurate to only the expected order of $p+1$ and has a more restrictive CFL condition. Due to the predictor-corrector formulation that does not involve methods of characteristics, these schemes extend easily to systems of equations in multiple dimensions. Concerning the case of two spatial dimensions (2D), for an advection using a Cartesian grid, when the flow does not align with the axes, especially when it is along a diagonal direction, the CFL conditions for the moment schemes also become restrictive and need improvement as will be shown by Fourier (von Neumann) analyses.

Introduction

The discontinuous Galerkin (DG) method is among the most popular high-order numerical methods for solving the compressible Navier-Stokes equations. It was introduced for the neutron transport equation by Reed and Hill [37], analyzed by LaSaint and Raviart [28] and developed and made popular for fluid dynamics equations by Cockburn and Shu [6], Bassi and Rebay [2,3], and others [4]. Efficient DG schemes using nodal points can be found in [16].

As opposed to the DG method, which employs the integral form, the flux reconstruction (FR) method introduced by this author [20,21] employs the differential form. The FR approach results in new methods as well as recovers existing schemes such as DG, staggered mesh [27], spectral volume [49], and spectral difference [29,48]. Reviews of FR methods can be found in [24,47,50].

Discretization for the spatial derivatives can be carried out effectively by DG or FR. When employed with a time discretization, however, the resulting scheme typically has either a very small time step size in the case of an explicit method or a very large system of equations in the case of an implicit one. Efforts to improve time discretization have been proposed by several authors. They generally result in trade-offs.

Concerning implicit schemes, it was shown by LaSaint and Raviart [28] that discretizing an ordinary differential equation (ODE) using DG results in an implicit Runge-Kutta (RK) method. In addition, if the solution is approximated by a polynomial of degree $p$ in time, the scheme is accurate to order $2p+1$. Their proof employs a left Radau polynomial. It turns out that by using the right Radau quadrature in the DG formulation, the resulting implicit RK method reduces to the right Radau collocation or Radau IIA method [22,24]. Implicit space-time DG schemes for conservation laws have been studied by several authors (e.g., [1,35,42]). Among more recent efforts include the dual time stepping procedure for fully implicit schemes by Jameson [26] and Loppi et al. [31] as well as implicit methods by Vermeire and Vincent [45] and Wang and Yu [46].

Concerning explicit time stepping methods, which is the main focus here, the standard choice is Runge-Kutta, and the resulting scheme is referred to as RK-DG [4,5]. Optimal RK-DG methods with the same order in both space and time have time step sizes corresponding to a CFL condition proportional to $1/\left(2p+1\right)$. The fully explicit ADER-DG (Arbitrary order using derivatives) schemes [8,9] approximate time to the same order of accuracy as space via a one-step approach. These schemes extend the ADER finite volume approach developed by Toro and co-workers (e.g., [40]) to the DG framework. The ADER-DG methods, similar to the space–time expansion discontinuous Galerkin or STE-DG scheme discussed below [11,32] turns out to have a more restrictive CFL condition than RK-DG. A closely related explicit time stepping called Lax-Wendroff type was developed for DG in [36] and for FR in [33]. Characteristic based space-time methods, which improve the CFL conditions by splitting out the advective part, can be found in [7,34].

It is generally believed that with explicit time stepping, the reduction in time step size is inevitable for DG type schemes due to the additional degrees of freedom within each cell as the order increases (say, $p+1$ Gauss solution points in a cell). This belief turns out to be false at least in the case of one spatial dimension (1D) as will be shown: a CFL condition of 1 for explicit DG type schemes is possible to arbitrary order.

Another important aspect of time stepping methods concerns their performance for large scale calculations using massively parallel supercomputers or graphics processing units (GPU) as discussed at PyFR.org [51]. Here, the DG and FR spatial discretization is highly parallelizable, and this ease of parallelization is retained with an explicit time stepping method but not with an implicit one.

Explicit methods with a CFL condition of 1 in 1D have been devised in the area of finite volume. As early as 1977, van Leer presented a series of five schemes for advection with this property [43]. Among the five, scheme I is the least accurate but became the most popular since it can easily be extended to the case of systems of equations; it is generally referred to as the MUSCL scheme (Monotone Upstream Scheme for Conservation Laws). Scheme III, which can be considered as an explicit one-step piecewise linear DG method for advection, is third-order accurate. Concerning the more accurate methods such as schemes III and V, the problem is, as stated in [44], “When trying to extend these schemes beyond advection, viz., to a nonlinear hyperbolic system like the Euler equations, the first author ran into insuperable difficulties because the exact shift operator no longer applies, and he abandoned the idea”.

For Van Leer's scheme III, the difficulty of extension was overcome by this author [19] by replacing the exact shift operator with a space-time formulation and a successive update procedure for the moments (coefficients of the Legendre components of the solution). The resulting method is called the (upwind) moment scheme. The approach was further analyzed and applied to nonlinear hyperbolic equations by Suzuki and van Leer [39]. As briefly discussed in [20], the moment scheme can be extended to arbitrary order. Such an extension was developed later by Lo [30] and studied in combination with van Leer's recovery scheme for diffusion. A more comprehensive extension of moment schemes to arbitrary order, together with the 1D and 2D Fourier analyses, was presented in [23].

In this paper, the two explicit DG methods studied are the STE-DG [11,32] and the moment schemes [19,23,30], where the latter method has not been published in refereed journals. The two methods, introduced independently, are both based on a ‘predictor-corrector’ formulation. The predictor step is the same; it is a Cauchy-Kovalevsky (CK) procedure, which involves no interaction of the data among neighboring cells. The basic idea is that in each cell, if the spatial derivatives to degree $p$ of the solution are known, the time and mixed derivatives to the same degree can be calculated by repeatedly differentiating the original partial differential equation [15]. This step is sometimes referred to as the Hancock scheme [41]. In practice, this step can be obtained using RK time stepping of appropriate order with no data interaction among cells. The corrector step of the two methods also shares the same space-time integration formulation. The two methods differ, however, in the evaluation of the space-time volume integrals. For the STE-DG scheme, these integrals are carried out in a straightforward manner using the space-time expansion in the predictor step. For the moment scheme, they are carried out by a successive procedure where, for all intermediate time stages, first, the cell average solutions are updated; the results are then employed in the updating of the first moments (slopes of the solutions); these results, in turn, are employed in the updating of the second moments (coefficients of the quadratic Legendre polynomial components of the solution), etc. This successive procedure is made possible by the observation that the derivative of a Legendre polynomial can be expressed as a linear combination of Legendre polynomials of lower order. The result is that for 1D advection, the moment scheme recovers the solution by van Leer's scheme III for $p=1$, has a CFL condition of 1 for all $p$, and is accurate to order $2p+1$, i.e., it possesses the super accuracy property. The STE-DG method, on the other hand, is simpler, but has a more restrictive CFL condition, and is accurate to the expected order of only $p+1$. Thus, in 1D, compared with the CFL condition proportional to $1/\left(2p+1\right)$ of standard RK-DG schemes, where space and time are discretized to the same order, the moment schemes provide a significant improvement. For the case of 2D advection using a Cartesian grid, however, there is a reduction in CFL condition when the flow does not align with the axes, and the largest reduction occurs when the flow is along a diagonal direction. The reason is that, in the DG formulation, there is no direct communication between cells that share only a single grid vertex with no common cell edge.

Note that compared with standard RK, the STE-DG scheme has the advantage of local time stepping, i.e., each cell can have its own time step size. This property makes the scheme highly efficient in applications with strongly varying time scales and has been effectively implemented in a 3D adaptive parallel simulation code (personal communication by Professor Gassner). Conceptually, due to the same space-time formulation, local time stepping applies to the moment scheme as well. This area of research, however, is beyond the scope of this work.

Concerning further study, the moment scheme employs the same collocation points in time as the implicit Radau IIA scheme [13,14], which is stable for arbitrary time step size. Thus, the Radau IIA method might be employed together with ideas and techniques from these explicit DG schemes to improve time step size in multiple dimensions and to avoid large matrix inversions.

The focus of this paper is on conceptual relations and basic ideas. The paper is organized as follows. In §2, van Leer's scheme III is reviewed except that the derivation here is for arbitrary order, not just piecewise linear. The predictor-corrector DG formulation as well as the STE-DG and the moment schemes in 1D are presented in §3. Section 4 provides 1D Fourier (von Neumann) analysis. Preliminary 1D numerical results are shown in §5. Extension to the 2D case is discussed in §6, and 2D Fourier analysis is presented in §7. Finally, conclusions and discussion can be found in §8.