Operator Bounds and Time Step Conditions for the DG and Central DG Methods

Abstract

Discontinuous Galerkin (DG) and central DG methods are two related families of finite element methods. They can provide high order spatial discretizations that are often combined with explicit high order time discretizations to solve initial boundary value problems. In this context, it has been observed that the central DG method allows larger time steps, especially for schemes with high accuracy. In this paper, we estimate bounds for the DG and central DG spatial operators for the linear advection equation. Based on these estimates and Kreiss-Wu theory, we obtain time step conditions to ensure the numerical stability of the DG and central DG methods when the methods are combined with locally stable time discretizations. In particular, for a fixed time discretization, the time step allowed for the DG method is proportional to $h/k^2$, while the time step allowed for the central DG method is proportional to $h/k$, where $h$ is the spatial mesh size and $k>0$ is the polynomial degree of the discrete space of the spatial discretization. In addition, the analysis provides new insight into the role of a parameter in the central DG formulation. We verify our results numerically, and we also discuss extensions of our analysis to some related discretizations.

Appendices

Appendix 1

In this “Appendix”, we illustrate how to find $\tilde{C}_{1,k}$ for use in (10) as described in Sect. 4.1. More specifically, for any $k\ge 0$, we want to find $\tilde{C}_{1,k}$ such that

$$\begin{aligned} {\Vert p'\Vert }_{L^2([-1,1])}\le \tilde{C}_{1,k}k^2{\Vert p\Vert }_{L^2([-1,1])} \end{aligned}$$

for any $p\in P^k([-1,1])$. By inspection, we see that $\tilde{C}_{1,0}=0$, so we consider $k\ge 1$. We can represent any $p\in P^k([-1,1])$ uniquely as $p(x)=\sum _{j=0}^k\alpha _j\tilde{L}_j(x),$ where $\{\tilde{L}_j(x)\}_{j=0}^k$ are the Legendre polynomials normalized so that ${\Vert \tilde{L}_j\Vert }_{L^2([-1,1])}=1$. We let $\alpha =(\alpha _j)\in {\mathbb R}^{k+1}$ denote the coefficient vector, and we let $A_k=\left( a_{ij}\right) \in {\mathbb R}^{(k+1)\times (k+1)}$, where

$$\begin{aligned} a_{ij}=\int \limits _{-1}^1\frac{d^{}\tilde{L}_{i-1}}{dx^{}}\frac{d^{}\tilde{L}_{j-1}}{dx^{}}dx, \qquad i, j=1, \ldots , k+1. \end{aligned}$$

We note that $A_k$ is a symmetric positive semi-definite matrix, and, in addition, that

$$\begin{aligned} {\Vert p'\Vert }^2_{L^2([-1,1])}={\alpha }^T{A_k}{\alpha } , \qquad {\Vert p\Vert }^2_{L^2([-1,1])}={\alpha }^T{\alpha }. \end{aligned}$$

Accordingly, we see that

$$\begin{aligned} (\tilde{C}_{1,k}k^2)^2 =\max _{\begin{array}{c} p\in P^k([-1,1])\\ p\ne 0 \end{array}} \frac{{\Vert p'\Vert }^2_{L^2([-1,1])}}{{\Vert p\Vert }^2_{L^2([-1,1])}} =\max _{\begin{array}{c} \alpha \in {\mathbb R}^{k+1}\\ \alpha \ne 0 \end{array}} \frac{{\alpha }^T{A_k}{\alpha }}{{\alpha }^T{\alpha }} =\lambda _{\mathrm {max}}(A_k), \end{aligned}$$

where $\lambda _{\mathrm {max}}(A_k)\ge 0$ is the largest eigenvalue of $A_k$, so we have

$$\begin{aligned} \tilde{C}_{1,k}=\frac{\sqrt{\lambda _{\mathrm {max}}(A_k)}}{k^2}. \end{aligned}$$

The process of finding $\tilde{C}_{2,k}$, $\tilde{C}_{3,s,k}$, and $\tilde{C}_{4,s,k}$ for use in (11), (12), and (13), respectively, is similar.

Table 5 Values of $\Xi (k;\sigma )$, where $C_{3,s}$ and $C_{4,s}$ are replaced by $\tilde{C}_{3,s,k}$ and $\tilde{C}_{4,s,k}$, respectively, as described in Sect. 4.1 and “Appendix 1”

Appendix 2

To illustrate how the local mesh regularly will affect the estimate in (21) for the central DG spatial operator on general meshes, we consider a quasi-uniform mesh with

$$\begin{aligned} x_{j+\frac{1}{2}} = \left( (1+\sigma )j+\frac{1-(-1)^j}{2}(1-\sigma )\right) \frac{h}{2} \end{aligned}$$

for any $j$, and $\sigma \in (0,1]$. For such mesh, $\rho _{j+\frac{1}{2}}=\sigma$ for all $j$, and $\sigma =1$ corresponds to a uniform mesh. In Table 5, we report the value of

$$\begin{aligned} \Xi (k;\sigma ) :=\left( 2C_{3,\frac{1}{2}}+2C_{3,\star }\right) k+4C_{4,\frac{1}{2}}C_{4,\star }(k+1), \end{aligned}$$

(29)

for different $\sigma$ and $k$, where the constants $C_{3, s}$ and $C_{4,s}$ are replaced by $\tilde{C}_{3, s, k}$ and $\tilde{C}_{4, s, k}$ as described in Sect. 4.1 and “Appendix 1”. From the results, one can see that the value of $\Xi (k;\sigma )$ is not affected dramatically by the local mesh regularity.