1. Introduction
The discontinuous Galerkin (DG) method may be viewed as a hybrid approach combining the good features of two classical numerical discretization approaches, the finite-volume (FV) and finite-element methods, and exploiting the merits of both. The DG spatial discretization combined with Runge–Kutta time integration (RKDG method) provides a class of robust algorithms for solving conservation laws (Cockburn and Shu 1989; Cockburn 1997). Because of its computationally attractive features such as local and global conservation, high-order accuracy, high parallel efficiency (petascale capability), and geometric flexibility, the DG method is becoming increasingly popular in atmospheric modeling [for the spherical geometry application, see, e.g., Giraldo et al. (2002); Nair et al. (2005a); Läuter et al. (2008)]. A recent review by Nair et al. (2011) presents various DG applications in atmospheric science with an extensive list of references.
Even though the RKDG scheme has many desirable properties, for transport equations with strong shocks or contact discontinuities, a nonlinear limiter must be employed to suppress oscillations. There are extensive studies on limiters for the low-order finite-volume methods. The slope limiters used for FV schemes can be extended to the modal form of DG methods (Cockburn 1997), however, this drastically reduces the high-order accuracy of the DG scheme (Iskandarani et al. 2005; Krivodonova 2007). To maintain the properties of the DG scheme, the limiter used should not reduce the order of accuracy in the region where the solution is smooth. To address this issue, Qiu and Shu (2005b) have developed high-order limiters based on the Weighted Essentially Nonoscillatory (WENO) schemes, where the WENO-type nonoscillatory reconstruction technique serves as a limiter for the RKDG method. A major disadvantage of this type of limiter is its requirement for a wider halo region (stencil), which could potentially impede parallel efficiency. For example, a third-order DG scheme requires a 5 × 5 stencil (i.e., halo size of width 2 in each direction) to apply a consistent WENO limiter. Subsequently, Qiu and Shu (2005a) improved this deficiency by developing the Hermite-WENO (H-WENO) limiter, for which a more compact stencil is employed. For a third-order DG scheme, the H-WENO limiter requires only a 3 × 3 stencil. The DG scheme combined with an H-WENO limiter has been recently used for a system of conservation laws in several applications (Luo et al. 2007; Balsara et al. 2007). Note that a WENO or H-WENO limiter is only “essentially” nonoscillatory by design, which implies that minor wiggles may appear in the solution even after the limiter is applied to the DG scheme. Therefore, the terminology “nonoscillatory DG scheme” is used throughout the text in a weak sense.
Though the WENO-type limiters can remove spurious oscillations, there is no guarantee that they will always keep the numerical solution within the physical bounds. The numerical solution may still have small-amplitude oscillations even after the limiter is applied. In other words, these schemes are not strictly positivity preserving. For many atmospheric tracers such as humidity and mixing ratios, the global maximum and minimum values are known in advance, and such tracers have “zero tolerance” for negative values. Therefore, for tracer transport models, the positivity preservation is considered to be a basic requirement. Very recently, Zhang and Shu (2010) developed a genuinely high-order bound-preserving (BP) filter for multidimensional RKDG methods, based on the Liu and Osher (1996) one-dimensional limiter. The BP filter clips extrema of the solution that go out of the physically legitimate bounds without violating the conservation property. Two nice features of the BP filter are that it is local and that it can easily be turned into a positivity-preserving filter when the lower bound is specified as zero. Zhang and Shu (2011) further extended this maximum-principle-satisfying filter to a variety of problems including systems of equations such as the Euler and shallow-water equations. The nonoscillatory and positivity-preserving properties for the proposed DG transport scheme are achieved by applying the H-WENO limiter and BP filter, respectively.
In this paper, we first introduce the basic DG scheme, with emphasis on a P2 modal (third order) version, and the implementation of an H-WENO limiter with a BP filter option. The basic ideas are developed in 2D Cartesian geometry and then extended to the cubed sphere (2D curvilinear) with several benchmark tests. The remainder of the paper is organized as follows: in section 2, the modal third-order DG scheme is described and the corresponding H-WENO scheme is introduced. Section 3 describes the implementation of the nonoscillatory DG scheme on the cubed-sphere geometry. In section 4, numerical tests on both a 2D Cartesian domain and the sphere are presented to demonstrate the performance of the nonoscillatory scheme, followed by some discussion and concluding remarks in section 5.
2. Nonoscillatory DG transport scheme
a. A third-order modal formulation
The integral equation (2) is the crux of the DG algorithm, the accuracy and efficiency of which are determined by the particular choice of Pk and the quadrature rules for the integrals. An arbitrary high-order modal DG discretization on the cubed sphere is given in Nair et al. (2005b). Nevertheless, our focus here is the development of a third-order modal nonoscillatory DG scheme. Details of the “modal” and “nodal” variants of DG scheme and their relative merits can be found in Nair et al. (2011).
A major limitation of the DG scheme is the stringent Courant–Friedrichs–Lewy (CFL) stability constraint associated with explicit time stepping. For high-order DG schemes employing polynomials of degree k > 1, an approximate CFL limit estimate is 1/(2k + 1) (Cockburn 1997). This is due to the fact that the DG method has more degrees of freedom (dofs) to evolve in time per element than a typical finite-volume or finite-element method; this makes the scheme highly accurate at a higher computational cost. However, reducing the order accuracy has some benefits. It significantly improves CFL stability and allows one to implement limiting algorithms based on those designed for FV methods. A moderate-order DG scheme, such as a third-order one (k = 2), has a CFL limit (approximately 0.2) comparable to some high-order FV schemes (Chen and Xiao 2008). In certain cases, a third-order DG scheme provides a solution that is qualitatively comparable to that of a fourth-order or fifth-order WENO scheme. Therefore, in this paper our main focus is the development of a practical third-order DG scheme, which we will refer to as the “P2-DG” scheme. For that we utilize the modal version of the DG method, which is amenable to limiting methods as used for FV schemes.
For the DG discretization, every rectangular element is mapped onto a standard element [−1, 1]2 with the local (ξ, η) coordinates. (a) GLL quadrature grid with 4 × 4 points, where on the boundary the solution points and flux points coincide. (b) GL quadrature grid with 3 × 3 internal points (filled circles) and three flux points on each side (filled squares). The values at the flux points are interpolated using the modal basis functions.
Citation: Monthly Weather Review 140, 9; 10.1175/MWR-D-11-00287.1
b. The H-WENO limiter
To suppress the oscillations in the numerical solution in the presence of a shock or discontinuity, we employ the Hermite-WENO limiting strategy. The H-WENO limiting strategy for the DG scheme was first introduced by Qiu and Shu (2005a). It is a variant of the traditional WENO-type limiter (Qiu and Shu 2005b) in the sense that they share the same methodology. Once a cell is identified as an oscillatory (or “troubled”) cell, with a shock detection technique such as the (total variation bounded) TVB-type limiter (Cockburn et al. 1990), then all the higher-order moments are modified except for the first moment (i.e., cell average). Preservation of the cell average is required to guarantee conservation. We briefly outline the procedure below (details can be found in Shu 1997).
The first step is to reconstruct several polynomials Pn using the information from the neighboring cells, for which a family of candidate stencils are required. Each big stencil contains several small stencils (or substencils) Sn consisting of the cell Ii,j and its neighbors, as depicted schematically in Fig. 2. The WENO scheme uses a convex combination of nonlinear weights wn from each stencil, which depends on the local smoothness of the solution, and ultimately creates the nonoscillatory solution. The smoothness indicators βn, which are a measure of the smoothness of the solution, are computed for each stencil; note that a smaller value of βn indicates a smoother function Pn in Sn. The smoothness indicators are then used to convert precomputed linear weights (γn) to nonlinear weights (wn). As a limiter, WENO modifies the high-order moments of the DG solution using a combination of nonlinear weights (Qiu and Shu 2005b).
The stencils used for the H-WENO reconstruction to limit a cell Ii,j are shown for a single index Iℓ=5. For the P2-DG scheme, the H-WENO limiter requires a big stencil with 3 × 3 cells, and eight small stencils S1, S2, … , S8. Each small stencil contains a combination of cells from the big stencil, including the cell to be limited (I5).
Citation: Monthly Weather Review 140, 9; 10.1175/MWR-D-11-00287.1
The major difference between the H-WENO and WENO scheme is that the former uses a more compact stencil than the WENO scheme for a given order of accuracy. This makes the H-WENO algorithm more desirable for a parallel computing environment. Unlike the WENO method, which employs only the cell averages from the neighboring cells, the H-WENO scheme relies on cell averages as well as derivative information (or “Hermite” information). For the DG discretization, derivatives are readily available, making the H-WENO limiter more computationally attractive. The H-WENO limiter retains all the nice qualities of the WENO limiter such as conservation and the nonoscillatory property, and it will not reduce the order of accuracy of the underlying scheme in the smooth region. For instance, the H-WENO stencil adopted for our P2-DG scheme uses a 3 × 3 “big” stencil, which is labeled in Fig. 2, as compared to the 5 × 5 stencil usually required by the WENO limiter.
Qiu and Shu (2005a) have shown that the H-WENO limiter coupled with the DG scheme is indeed third-order accurate for smooth problems. However, just like the WENO limiter, the H-WENO limiter is also only essentially nonoscillatory, which means it may not eliminate all small oscillations near the physical bounds. This is the motivation for us to further implement a bound-preserving filter for the P2-DG scheme combined with the H-WENO limiter.
c. The BP filter
To preserve the initial bounds of the numerical solutions and eliminate negative densities (if positivity is a requirement), we can apply a bound-preserving filter as an additional option. The BP filter has several attractive features. It is local, conservative, computationally cheap, and easy to implement (Zhang and Shu 2010). For P2-DG, the moments are evolving with respect to time. However, the gridpoint values of the approximate solution Uh on Ii,j at any instant can be computed from the polynomial representation (4). Let pi,j(x, y) be the modal DG polynomial on the cell Ii,j with cell average 
The filter (12) is in fact based on the 1D filter developed by Liu and Osher (1996) for an FV scheme. Zhang and Shu (2010) have proved that this filter satisfies the strict maximum principle and is genuinely high order and extensible to DG applications. We note that, although the BP filter keeps the bounds of the solution in the range [m*, M], there is no guarantee that it will remove all the internal oscillations within the cells. However, the combination of the H-WENO limiter and the local BP filter addresses this issue.
3. Nonoscillatory DG scheme for cubed sphere
The cubed-sphere geometry (Sadourny 1972) has become a popular choice for the spherical grid system in global modeling because it offers a quasi-uniform rectangular grid structure on the sphere without pole problems. This grid structure is suitable for high-order element-based Galerkin methods as well as cell-centered FV methods. There are different variants of the cubed-sphere topology, but we will consider the cubed-sphere geometry employing the equiangular central projection as described in Nair et al. (2005b). The sphere 
4. Numerical experiments
a. Cartesian tests
To validate the nonoscillatory DG scheme, we solve (1) without a source term on Cartesian domains using several benchmark tests, including a solid-body rotation test suggested by LeVeque (2002). The initial scalar fields include smooth and nonsmooth distributions to check the effectiveness of the limiter and filter. For the following tests we use the 3 × 3 GL quadrature grid shown in the right panel of Fig. 2, and the third-order SSP Runge–Kutta (Gottlieb et al. 2001) method for time integration.
1) Advection of a 1D irregular signal
To see the effects of the BP filter and H-WENO limiter on the P2-DG scheme, we first solve a simple one-dimensional form of the conservation law (1), ψt + (uψ)x = 0, on a periodic domain [−π, π]. The initial value of the scalar ψ(x, t = 0) = ψ0 is given as a three-level step function (or irregular signal) in [0, 1], with values of ψ0 = 0, 0.5, 1 for x ∈ [−π, π], as indicated by thin lines in Fig. 3. The computational domain consists of 80 cells with a uniform velocity of u =1. In Fig. 3, the numerical solution ψn after one revolution (period) is shown as dashed lines. Although the BP filter keeps the numerical solution within the initial bounds [0, 1] as shown in Fig. 3b, the solution remains oscillatory at the level ψn = 0.5, similar to the P2-DG case (Fig. 3a). However, the H-WENO limiter with the BP filter removes the oscillations (Fig. 3c) while strictly preserving the initial bounds. The P2-DG solution combined with the H-WENO limiter appears to be very similar to Fig. 3c (not shown), but in this case the solution is not strictly positivity preserving, as there are minor undershoots (overshoots) at ψn = 0. In other words, the purpose of the BP filter is to preserve the solution within the initial bounds; nevertheless, it has no control over the internal oscillations of the solution or it cannot make the solution nonoscillatory. In general, the P2-DG solution with the H-WENO and BP combination is essentially nonoscillatory, but preserves the bounds of the initial data.
Results after one complete period for the uniform advection of an irregular signal (three-level step function) on a 1D periodic domain [−π, π] employing 80 cells. Reference solution is marked as thin solid lines and numerical solution is marked as dashed lines. Numerical solution with (a) P2-DG, (b) P2-DG and BP filter, and (c) P2-DG with both H-WENO limiter and BP filter.
Citation: Monthly Weather Review 140, 9; 10.1175/MWR-D-11-00287.1
2) Solid-body rotation of a Gaussian hill
Convergence plots for 2D Cartesian solid-body rotation of a Gaussian hill. The P2-DG transport scheme combined with the H-WENO limiter or BP filter or both is used for the convergence tests.
Citation: Monthly Weather Review 140, 9; 10.1175/MWR-D-11-00287.1
3) Solid-body rotation of a nonsmooth distribution
Figure 5 shows the contour plots of the numerical results and the exact solution after finishing a full circle of rotation. The grid resolution is 80 × 80, which means Δx = Δy = 0.025, on a square domain [−1, 1]2. In Fig. 6a, a 3D projection of the numerical solution is shown. It can be seen clearly from the results that the H-WENO limiter effectively removes the nonphysical oscillations that occur near the discontinuities when using a DG scheme. Figures 6a–c show a 3D projection of the solution on a 64 × 64 mesh with the DG scheme without the limiter or filter, with the BP filter, and with both the H-WENO limiter and BP filter, respectively. The BP filter alone helps to keep the numerical solution within the physical bounds [0, 1] as shown in Fig. 6b, but there may be oscillations within these bounds. However, the H-WENO and BP combinations completely eliminate oscillations while being strictly positive definite, as evident from Fig. 6c. Thus, using the nonoscillatory DG scheme achieves a substantial improvement in the quality of the numerical solution over using the DG scheme alone.
Numerical results of solid-body rotation of a nonsmooth distribution (a circular cone and a square block) after one revolution with Δx = Δy = 0.025 using the (a) P2-DG scheme only, (b) P2-DG with the BP filter, (c) P2-DG with the H-WENO limiter, and (d) P2-DG with the H-WENO limiter and BP filter. Solid lines show contour values from 0.05 to 0.95 with increments of 0.1. Thick solid lines indicate the contour values of 0.05 and 0.75 of the exact (reference) solution.
Citation: Monthly Weather Review 140, 9; 10.1175/MWR-D-11-00287.1
A 3D perspective of the numerical solution shown in Fig. 5 but on a 64 × 64 grid. (a) P2-DG oscillatory solution, (b) P2-DG with the BP filter, where the bounds are preserved but oscillations are visible within the bounds [0, 1], and (c) a nonoscillatory solution using the P2-DG scheme combined with the H-WENO limiter and the BP filter.
Citation: Monthly Weather Review 140, 9; 10.1175/MWR-D-11-00287.1
b. 2D spherical tests
1) Solid-body rotation: Cosine bell
In Table 1, we give normalized standard error norms at t = T = 12 days. Normalized standard error measures are comparable to those seen in the recent high-order FV models (Chen and Xiao 2008; Ullrich et al. 2010; Katta et al. 2012, manuscript submitted to Quart. J. Roy. Meteor. Soc.). Figure 7 shows the contour plots of the numerical solution after one full rotation. From the results, it is clear that the nonoscillatory scheme will eliminate the negative values produced by the DG scheme near the foot of the bell.
Normalized standard errors for ψ for the solid-body (cosine bell) rotation test after a full revolution. The third-order P2-DG transport scheme with different limiter (filter) combinations is used for the test. The flow orientation is along the northeast direction (α = π/4) on a 32 × 32 × 6 cubed-sphere mesh with a time step Δt = 600 s. Minimum and maximum heights of the cosine bell (ψ) after a revolution are indicated by Min ht and Max ht, respectively.
An orthographic projection of the solution of the solid-body rotation of a cosine bell on the cubed sphere. The wind fields are oriented along the northeast direction (α = π/4), and 12 days are required for a full revolution. (a) Initial height of the cosine bell, which ranges from 0 to 1000 units, (b) P2-DG solution where spurious undershoots can be seen, (c) numerical solution with the H-WENO limiter and BP filter, and (d) numerical solution with the BP filter. The cubed sphere with a mesh of 32 × 32 × 6 and a time step Δt = 600 s are used for the simulation.
Citation: Monthly Weather Review 140, 9; 10.1175/MWR-D-11-00287.1
2) Solid-body rotation: Multiscale signal
Figure 8a shows the initial condition for the multiscale signal (18), and Fig. 8b shows the numerical solution (after 12 days) with the P2-DG scheme and the H-WENO limiter. The experimental setup is similar to that of the solid-body rotation test, where a mesh with 32 × 32 × 6 cells and a time step of Δt = 600 s are used. The P2-DG numerical solution without the H-WENO limiter is visually indistinguishable from that with the H-WENO limiter (Fig. 8b) and therefore not shown. Table 2 lists the normalized errors for the P2-DG scheme with or without the H-WENO limiter. Figures 8c,d show the value of ψ(λ, θ) sampled along the equator (1D data) after one revolution using the P2-DG scheme without and with the H-WENO limiter, respectively. The exact solution is indicated using black dots and the numerical solution is displayed as a solid line. For this test, it is clear that the P2-DG scheme itself is capable of handling the shock in the multiscale signal, and the H-WENO limiter has only a marginal impact on the solution (see Fig. 8d; Table 2). Moreover, it is found that the BP filter has little or no influence on the solution, as is evident from Table 2.
Solid-body rotation test of a multiscale signal comprising continuous and discontinuous functions: (a) exact (initial) solution and (b) numerical solution after a revolution (12 days) with P2-DG and H-WENO; for this test, the wind field is oriented along the equator (α = 0). The 1D numerical solution (solid lines) sampled along the equator after 12 days, and the exact reference solution marked as thick black dots: (c) P2-DG solution and (d) solution with P2-DG and H-WENO combination. The cubed-sphere mesh with 32 × 32 × 6 cells and a time step Δt = 1440 s are used for the numerical simulations.
Citation: Monthly Weather Review 140, 9; 10.1175/MWR-D-11-00287.1
Normalized standard errors for ψ for the solid-body rotation test with a multiscale signal on a 32 × 32 × 6 cubed-sphere mesh. The third-order P2-DG transport scheme with H-WENO limiter and/or BP filter combinations is used for the test. The standard global error measures are based on Williamson et al. (1992).
3) Deformational flow on the sphere: Vortex problem
We consider a special case of the deformational flow test, the moving-vortex problem, introduced in Nair and Jablonowski (2008). This test consists of two steady vortices, which are created on a sphere and whose centers are located at diametrically opposite sides. The flow field is nondivergent, time-dependent, and highly deformational. The vortices are designed to move along a great-circle trajectory while deforming, and the analytic solution is known at a given time. However, we use a static option for the vortices so that the vortex centers remain at the initial position and are numerically integrated for an extended period of time (60 days) as opposed to the recommended 12 days (Nair and Jablonowski 2008). The purpose of this test is to check the vortex filament formation at the smallest resolvable scale by the numerical model (Pudykiewicz 2011; Flyer and Wright 2007).
The numerical experiment is performed with a relatively high-resolution mesh employing 100 × 100 × 6 cells and a time step of Δt = 600 s. Figure 9 shows the solution after 36 and 60 model days. The initial condition 
The solutions for the deformational flow (vortex) test on the cubed sphere at simulated days 36 and 60: (a) initial solution; and day 36 (b) P2-DG solution and (c) P2-DG solution with H-WENO. Day 60 (d) exact (reference) solution, (e) P2-DG solution, and (f) P2-DG solution with H-WENO. Note that the numerical solutions with or without the BP filter are visually indistinguishable. The cubed-sphere mesh with 100 × 100 × 6 cells and a time step Δt = 600 s are used for the numerical simulations.
Citation: Monthly Weather Review 140, 9; 10.1175/MWR-D-11-00287.1
Note that the deformational (vortex) test is a smooth problem for which the DG scheme does not require a limiter. This is also clear from the multiscale signal test (a quasi-smooth case) considered above. The P2-DG scheme is local and relies only on the cell in question, and even with the application of the BP filter, its local data dependency does not change. However, when the H-WENO limiter is applied alone with the P2-DG scheme, sharp gradients (thin filaments) in the solution are smoothed out because of excessive limiting. This is also due to the fact that the H-WENO limiter depends on a 3 × 3 wide stencil. If minor oscillations are present in a cell the limiter may be activated, employing the values from the least oscillatory cells in the stencil. This often leads to the flattening of legitimate sharp peaks, similar to the effect of a slope limiter. A way to avoid unwanted limiting (excessive dissipation) by the H-WENO limiter is to employ better (stringent) criteria for identifying oscillatory (troubled) cells.
We have also computed the normalized standard l1, l2, and l∞ errors after 12 model days. These values, for the P2-DG case with and without the BP filter, are virtually identical. When approximated to two decimal places, they are 6.93 × 10−6, 3.30 × 10−5, and 8.91 × 10−4, respectively. The corresponding values for the P2-DG case combined with the H-WENO and BP combination are 1.31 × 10−5, 6.81 × 10−5, and 1.61 × 10−3, respectively. The ψmax for all the cases is very close and approximately equal to 8.77 × 10−9. The ψmin for the P2-DG case is 1.58 × 10−7 and for the other two cases it is approximately equal to 4.04 × 10−8.
4) Deformational flow on the sphere: Slotted cylinder
Numerical solution for the deformational flow test on the cubed sphere with nonsmooth (twin slotted cylinder) initial conditions. (a) The initial slotted cylinders move along the zonal direction while deforming, and return to the initial position after making a complete revolution. The mesh size is 45 × 45 × 6 and Δt = 0.001 25. The P2-DG solution at halftime (t = T/2) with the (b) BP filter and H-WENO combination and (c) BP filter, respectively. (d) The slotted cylinders after one cycle of revolution (deformation) with the BP and H-WENO combination at the final time T = 5.
Citation: Monthly Weather Review 140, 9; 10.1175/MWR-D-11-00287.1
Figure 10 shows the results of the deformational flow tests with the P2-DG scheme combined with different limiter/filter options. Figures 10c,d show the results for the slotted-cylinder case at the halftime (t = T/2) and final time (t = T). The P2-DG scheme combined with the H-WENO limiter and BP filter at resolution 45 × 45 × 6 captures the original shape of the slotted-cylinder as shown in Fig. 10d. The P2-DG scheme with the BP filter produced visibly identical results (not shown). For brevity, we do not show the deformational results with the twin cosine-bell problem, but error norms are tabulated in Table 3.
Normalized standard errors for ψ for the deformational flow test with the twin SC and twin CB cases on a 45 × 45 × 6 mesh. The third-order P2-DG transport scheme with different limiter (filter) combinations is used for the test.
As shown in Fig. 10, the nonoscillatory DG scheme preserves the discontinuity quite well and completely removes undesirable overshoots and undershoots. In Table 3, the normalized standard errors are given for different combinations of the limiter (filter) for the nonsmooth case. The l1, l2, and l∞ errors for the nonsmooth case are significantly higher than those shown for the quasi-smooth case in Table 3. This is because the initial data for the slotted-cylinder case are severely nonsmooth (C0 discontinuous). However, for the twin cosine-bell case (Table 3), the results compare well with those reported in the Nair and Lauritzen (2010) paper.
We provide a rough estimate of the additional computational overhead required for the BP filter and H-WENO limiter. Both filter and limiter are applied at every stage of the third-order Runge–Kutta (RK3) time stepping scheme. As compared with the DG (oscillatory) scheme, it is found that the DG scheme and BP filter combination takes 28% more computational time, while the DG scheme combined with the H-WENO limiter and BP filter consumes about 40% more time. Note that the H-WENO scheme is selectively applied only to the cells that need limiting, while the BP filter is applied to each cell. Also, we notice that the BP filter could be applied at the last stage of the RK3 time stepping without incurring any significant change in the quality of solutions. This might be an efficient option for some application where the passive tracer transport is performed in isolation from the model dynamics.
We have also tested the effectiveness of the BP limiter in a nodal version of the DG scheme as described in Nair and Lauritzen (2010) (fourth order using 4 × 4 GLL points). Implementation of the BP scheme in the nodal DG version is straightforward because the solution evolves in physical (gridpoint) space. It is found that the numerical solution for the nonsmooth case is very similar to those shown in Fig. 10. However, for fifth- and higher-order nodal DG versions, the solutions are still within the physical bounds, but increasingly contaminated by internal oscillations within the elements.
5. Summary and conclusions
The discontinuous Galerkin (DG) methods are not inherently nonoscillatory. When there are discontinuities and sharp gradients in the solution, the DG transport schemes generate spurious oscillations that are unacceptable for many practical applications. The main focus of this paper is the development of a third-order DG transport scheme that is amenable to limiting processes and has more lenient CFL stability when used with explicit time stepping. A third-order modal version of the DG scheme (P2-DG) with 6 degrees of freedom per element (cell) was developed in Cartesian geometry. To suppress oscillations, a limiter based on the Hermite-Weighted Essentially Nonoscillatory (H-WENO) method was applied to the P2-DG scheme. The H-WENO limiter uses a 3 × 3 computational stencil such that the oscillatory cell is located at the center. Although it suppressed oscillations, the H-WENO limiter cannot guarantee that the legitimate physical bounds of the initial solution are maintained, and oscillations of very small amplitude may still remain in the solution. To address this issue, a bound-preserving (BP) conservative filter was combined with the H-WENO limiter. The BP filter is local to the element and computationally efficient. This option provides strict positivity preservation for the P2-DG scheme. An explicit third-order Runge–Kutta (RK3) method was adopted for time integration.
To validate and verify the resulting nonoscillatory DG scheme, a variety of benchmark tests were performed. The H-WENO limiter and BP filter were optionally applied to remove the nonphysical oscillations and to keep the numerical solution within the physical bounds. The effects of the H-WENO limiter and BP filter on the P2-DG scheme were demonstrated using a simple 1D test. On the 2D Cartesian domain, two solid-body rotation tests were used. With Gaussian initial data for solid-body rotation, the test shows that the nonoscillatory scheme is indeed third order. The solid-body rotation with nonsmooth data shows that the scheme is nonoscillatory and positivity (bound) preserving. However, there is a slight degradation with the H-WENO limited solution. This could be due to “excessive limiting” on the cells that do not require limiting. A better limiting criterion, other than the TVB shock detection method currently used for the H-WENO scheme, might improve the H-WENO solution further.
The nonoscillatory P2-DG scheme was then extended to the spherical (cubed sphere) geometry. On the cubed sphere, a standard advection (cosine bell) test was performed first to test the nonoscillatory scheme. In addition, solid-body rotation of a multiscale signal that contained smooth and nonsmooth regions was considered. Results show that the nonoscillatory scheme eliminates all negative values and small oscillations, which occur near the foot of the cosine bell in the DG solution, without affecting the accuracy of the DG scheme. On the sphere, two deformational flow tests (with both smooth and nonsmooth fields) were used to further validate the P2-DG scheme with the H-WENO limiter and the BP filter. For the smoothly deforming vortex problem, the P2-DG scheme could easily resolve the fine filament structures of the vortex on a 100 × 100 × 6 mesh after 60 model days. However, the solution with the H-WENO limiter could not resolve the fine filaments because of excessive limiting as seen in the Cartesian case. For this test, it is found that the BP filter has only a very small impact on the fine filament structure. This indicates that the P2-DG scheme does not need limiting for smooth problems. If positivity preservation is an issue then the BP filter is a very good choice, and the filter does not adversely affect the quality of the DG solution.
A new challenging deformational flow test was also used to assess the performance of the nonoscillatory scheme in the presence of discontinuities. For this test, nonsmooth (slotted cylinders) and quasi-smooth (cosine bells) initial data were used. Numerical results, which were obtained on a 45 × 45 × 6 mesh, show that the nonoscillatory scheme provides a good approximation to the exact solution. The standard relative errors, and the global maximum and minimum, show that there is a substantial improvement in the quality of the solution of the nonoscillatory scheme as compared to the solution obtained by the regular DG scheme. Extending the P2-DG transport scheme to the 3D case is straightforward at a higher computational cost. Nevertheless, a dimension-split approach might save computational expenses significantly and ease the implementation of the limiter.
Acknowledgments
The authors are thankful for Dr. Richard Loft (NCAR) and Professor Henry Tufo (CU Boulder) for the SIParCS internship support for Yifan Zhang, and Prof. C.-W. Shu (Brown University) for the inspiration. The authors gratefully acknowledge the internal review by Dr. David Hall (NCAR), and would like to thank three anonymous reviewers for their constructive comments. This project is supported by the DOE BER Program DE-SC0001658.
APPENDIX A
H-WENO Reconstructions
In section 2, we discussed the P2-DG discretization for each cell Ii,j = Iℓ(i,j), where we considered local coordinates (ξ, η) on Iℓ, and the orthogonal basis set 
- On stencil S1, we require P1(x, y) to satisfyUsing the above six constraints, P1 is reconstructed as
- On stencil S2, we require P2(x, y) to satisfy
- On stencil S3, we require P3(x, y) to satisfy
- On stencil S4, we require P4(x, y) to satisfy
- On stencil S5, we require P5(x, y) to satisfy
- On stencil S6, we require P6(x, y) to satisfy
- On stencil S7, we require P7(x, y) to satisfy
- On stencil S8, we require P8(x, y) to satisfy
APPENDIX B
The Smoothness Indicator
- For the mode
For each Pn(x, y), this gives - For the mode
For each Pn(x, y), - For the higher-order modes,
For each Pn(x, y),
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