Stable Filtering Procedures for Nodal Discontinuous Galerkin Methods

High frequency errors are always present in numerical simulations due to inaccuracies of the spatial derivatives at high-wave numbers. A common technique to combat such errors is to use a filter operator that removes high wave number oscillations, e.g. [7, Sect. 5.3]. The filtering procedure is separate from the scheme itself, and is often done in an ad hoc fashion. Filtering is applied “as little as possible, but as much as needed”.

Recently, Lundquist and Nordström [11] removed the ad hoc nature of filtering in finite difference (FD) methods. They derive a contractivity condition on the explicit filter matrix by re-framing the filtering procedure as a transmission problem [13]. The work in [11] develops a necessary condition for the filter matrix and how it must be related to the particular discrete quadrature rule.

The goal of the present work is to remove the ad hoc nature of the nodal DG filtering procedure and prove that the commonly used explicit filter technique from the spectral community [5, 6] is stable. Just as in the case of FD methods, the stability of the filtering for nodal DG relies upon the existence of a high-accuracy auxiliary filter matrix as well as a semi-discrete bound on the solution.

The remainder of the paper is organized as follows: Sect. 2 gives an overview of the nodal DG method and the commonly used filtering procedure. Section 3 generalizes the theoretical stability results from [11] into the DG context. Numerical results that verify the theoretical findings are found in Sect. 4. Concluding remarks are given in Sect. 5.

DG methods are principally designed to approximate solutions of hyperbolic conservation laws, see [7, Chap. 2] or [9, Sect. 4.8] for details. Here, we consider such a conservation law in one spatial dimensionwhere \(u\equiv u(x,t)\) is the solution and \(f\equiv f(u)\) is the flux function. The conservation law is then equipped with an initial condition \(u(x,0)\equiv u_{\text {ini}}(x)\) and boundary condition(s).

To arrive at the nodal DG approximation we first transform the domain \([x_L,x_R]\) to the reference interval \([-1,1]\). To do so, we apply an affine mapping \(\xi (x)\in [-1,1]\) [9, Eq. 8.10] and rewrite the conservation law in the computational coordinate \(\xi \). The nodal DG approximation is built from the weak form of the mapped equation. The solution and fluxes are approximated by Lagrange polynomials of degree N and the inner products in the variational form are approximated with high-order Legendre-Gauss-Lobatto (LGL) quadrature. Full details are given in [9, Sect. 4.7].

The resulting strong form, nodal DG approximation is thenwhere \(\underline{U} = \left[ U_0,U_1,\ldots ,U_N\right] ^T\) is the vector form for the degrees of freedom and \(\underline{F}\) is the discrete flux. The matrices in (2.1) are the discrete derivative matrix [9, Sect. 3.5.2] \(\mathcal {D}_{ij}=\ell '_j(\xi _i)\) where \(\{\ell _j(\xi )\}_{j=0}^N\) are the nodal Lagrange basis polynomials and \(\{\xi _i\}_{i=0}^N\) are the LGL quadrature nodes, the mass matrix \(\mathcal {M}= \text {diag}(\omega _0,\ldots ,\omega _N)\) containing the LGL quadrature weights and the boundary matrix \(\mathcal {B}= \text {diag}(-1,0,\ldots ,0,1)\). There exist several numerical flux functions depending on the particular mathematical flux function f(u), see Toro [16, Chap. 2] for details. The numerical flux used in this work is the Lax-Friedrichs flux, \(F^* = (F^R+ F^L - \lambda _{\max }(U^R-U^L))/2\) where \(\lambda _{\max }\) is the largest wave speed, see [4, Sect. 4.2] or [16, Sect. 5.3.4] for details.

Two features to note for this flavour of nodal DG approximation are: Firstly, the diagonal mass matrix \(\mathcal {M}\) denotes a quadrature rule that is exact for polynomials up to degree \(2N-1\). So, there is equality between the continuous and discrete integralprovided the product of the functions v and w is a polynomial of degree \(\le 2N-1\). From (2.2), the continuous and discrete \(L_2\) norms are \(\left\langle v,v\right\rangle = \left| \left| v\right| \right| ^2\) and \(\left\langle V,V\right\rangle _{\!N} = \left| \left| V\right| \right| _{N}^2\). Secondly, the mass and derivative matrices of the LGL collocation DG scheme (2.1) form a summation-by-parts (SBP) operator pair [14] for any nodal polynomial order N [4, Sect. 2.1], i.e. \(\mathcal {M}\,\mathcal {D}+ (\mathcal {M}\,\mathcal {D})^T = \mathcal {B}\). From the SBP property, stable versions of the nodal DG method can be constructed [3, 4].

Filtering is separate from the spatial discretization and changes the approximate solution during the time integration procedure. To discuss the filtering and its effect on stability we follow [11] into the nodal DG context. With homogeneous boundary conditions, semi-discrete stability ensures that the discrete norm of the approximate solution is bounded by the discrete norm of the initial conditions, see [12, Sect. 4] for complete details. For the nodal DG approximation such a stability statement takes the form \(\left| \left| U(t)\right| \right| _{N} \le \left| \left| U_{\text {ini}}\right| \right| _{N}\), where \(U_{\text {ini}}\) is the initial condition evaluated at the LGL nodes.

Pursuant to the work [11], we view the application of a filter matrix to a discrete solution at some intermediate time \(t_1\) as a transmission problem [13, Sect. 3]:where the operator \(\mathbb {D}\) contains the derivative matrix \(\mathcal {D}\) as well as the boundary conditions. The filtering stated in the final term of (3.1) is performed in an explicit fashion. For stability it must hold that the filter is contractive, i.e. \(\left| \left| V(t_1)\right| \right| _{N} \le \left| \left| U(t_1)\right| \right| _{N}.\) In turn, this contractive property guarantees that the filter procedure is stable because \(\left| \left| V(t_1)\right| \right| _{N} \le \left| \left| U(t_1)\right| \right| _{N} \le \left| \left| U_{\text {ini}}\right| \right| _{N}\).

The contractivity property implies that the following contractivity condition [11, Eq. 7] on the explicit filter matrix \(\mathcal {F}\) must holdThe contractivity condition (3.2) expresses a precise interplay between the filter matrix and the mass matrix. A necessary condition for the explicit filter matrix to satisfy (3.2) is that an auxiliary filter matrix \(\widetilde{\mathcal {F}}_{\text {aux}}= \mathcal {M}^{-1}\mathcal {F}^T\mathcal {M}\), exists and possesses the same accuracy as \(\mathcal {F}\), otherwise (3.2) is provably indefinite [11, Sect. 4.1].

We will show that the auxiliary filter matrix is identical to the original filter matrix for the LGL nodal DG approximation. Furthermore, the filter matrix \(\mathcal {F}\) indeed satisfies the contractivity condition (3.2). Both results require the following Lemma.

We can now prove

The following result is then self-evident.

Further, the result from Lemma 1 allows us to prove

The discussions above are for a single spectral element. However, the contractive filter matrix \(\mathcal {F}\) and its stability carries through to a multi-element DG approximation. At a given time and applied locally within each element, the filtering in a DG formulation is stable. This is because the underlying spatial approximation is constructed to be stable, e.g. [4, 10]. Therefore, the overall discretization in a given time step is, for example: (1) Filter the current solution at time \(t_n\) in each element using the above described filter; (2) Apply the stable DG coupling of the elements by numerical flux functions (3) Evolve the solution in all elements one time step to \(t_{n+1}\); (4) Repeat the procedure from (1). Such an algorithm retains a semi-discrete stability estimate.

We consider two test problems that develop a non-smooth solution over time and select \(\alpha = 36\), \(N_c = 4\) and \(s = 16\) for the filter parameters in (2.3). To integrate the semi-discrete DG method (2.1) in time, we use a third-order, low storage Runge–Kutta scheme of Williamson [18, Table 1].

As a first example we consider a variable coefficient linear advection problem, \(u_t + a(x)u_x = 0\), with a bounded solution that develops steep gradients [6, Sect. 3.2]. The variable wave speed is \(a(x) = \sin (\pi x - 1)/\pi \) on the domain \(\Omega =[-1,1]\). The wave speed a(x) is positive at the boundaries of the domain. Therefore, no boundary condition is set at the right and at the left the boundary condition is enforced weakly through the numerical flux function and the prescription of an external solution state using the exact solution. We take the initial condition \(u_{\text {ini}}(x) = \sin (\pi x)\) with the corresponding analytical solution \(u(x,t) = \sin \left( 2\tan ^{-1}\left( e^{-t}\tan \left( (\pi x -1)/2\right) \right) + 1\right) \) [6, Sect. 3.2]. In Fig. 1 we compare the solutions at time \(T=4\) using polynomial order \(N=256\) and \(\Delta t = 1/2000\). The unfiltered DG scheme contains spurious oscillations whereas the filtered one does not. The filter matrix was applied to the solution after every time step in the simulation.

The second example illustrates the importance of the stability of the underlying spatial discretization. We consider Burgers’ equation and two forms of the nodal DG spatial discretization. One discretizes the conservative form of the PDE where the nonlinear Burgers’ flux is \(f(u) = u^2/2\) while the other writes the spatial derivative of the flux in a split formulation \(f_x(u) = ((u^2)_x + uu_x)/3\). On the continuous level these two forms are equivalent; however, on the discrete level they exhibit different behavior. Most notably, the solution energy \(u^2/2\) is bounded for the nodal skew-symmetric DG discretization whereas no such bound exists for the discrete conservative form [4, Sect. 4]. If the underlying numerical scheme is provably stable, and the filtering is contractive, it will remove energy from the solution in a stable controlled way. However, if the underlying scheme is unstable the filtering still removes energy and the approximation might seem stable.

To illustrate this we consider the domain \(\Omega =[0,2]\) with periodic boundary conditions and the initial condition \(u_{\text {ini}}(x) = \frac{1}{5}[1 + \cos (\pi x)]\). We run the simulation with polynomial order \(N=128\) up to \(T=2.25\) and filter the solution at 16 equally spaced times. We run four variants of the nodal DG scheme: Conservative formulation; unfiltered and filtered as well as skew-symmetric formulation; unfiltered and filtered. In Fig. 2a we present the evolution of the solution energy, normalized with its initial value. Figure 2b–d present the approximate solution at the final time produced by the filtered conservative, unfiltered skew-symmetric and filtered skew-symmetric DG schemes, respectively. The reference solution is created with a standard finite volume scheme on 10,000 grid cells.

The simulation is well resolved and the four variants are nearly indistinguishable for most of the simulation time. However, as the gradients steepen we see that the conservative formulation, for which no energy stability exists, behave erratically. The unfiltered conservative simulation crashes at \(T\approx 1.8\). The filtered conservative simulation runs as the filtering keeps the solution energy “under control.” In Fig. 2a we observe growth in the solution energy between the filter applications because the underlying spatial discretization is unstable. The solution energy of unfiltered and filtered skew-symmetric simulations both remain bounded because the underlying spatial discretization possesses an energy bound. Comparing Fig. 2b, d, we see at the final time the filtered conservative and skew-symmetric DG schemes produce nearly identical results. As previously noted, the unfiltered conservative DG scheme crashes while the skew-symmetric DG scheme runs. The solution energy \(u^2/2\) for Burgers’ equation is a conserved quantity for smooth solution and should only dissipate in the presence of discontinuous solutions [4, Sect. 4]. The skew-symmetric DG approximation is constructed to conserve the solution energy \(u^2/2\). So, in the absence of dissipation spurious oscillations develop near the discontinuity and propagate throughout the domain as energy is redistributed at the smallest resolvable scale [15, Sect. 6]. Thus, as shown in Fig. 2c, we see that the unfiltered skew-symmetric scheme is stable but the solution quality is extremely poor due to unphysical ringing.

We proved that the commonly used nodal DG filter matrix \(\mathcal {F}\) satisfied a contractivity condition. This result implied that the explicit filtering procedure for nodal DG methods is stable. Numerical results verified that the filtering was efficient and stable if the underlying spatial discretization of the method had a semi-discrete bound.