Superconvergence of discontinuous Galerkin and local discontinuous Galerkin methods: Eigen-structure analysis based on Fourier approach

Abstract

Various superconvergence properties of discontinuous Galerkin (DG) and local DG (LDG) methods for linear hyperbolic and parabolic equations have been investigated in the past. Due to these superconvergence properties, DG and LDG methods have been known to provide good wave resolution properties, especially for long time integrations (Zhong and Shu, 2011) [26]. In this paper, under the assumption of uniform mesh and via Fourier approach, we observe that the error of the DG or LDG solution can be decomposed into three parts: (1) dissipation and dispersion errors of the physically relevant eigenvalue; this part of error will grow linearly in time and is of order: $2k+1$ for DG method and $2k+2$ for LDG method (2) projection error: there exists a special projection of the exact solution such that the numerical solution is much closer to this special projection than the exact solution itself; this part of error will not grow in time (3) the dissipation of non-physically relevant eigenvectors; this part of error will be damped exponentially fast with respect to the spatial mesh size $\mathrm{\Delta }x$. Along this line, we analyze the error for a fully discrete Runge–Kutta (RK) DG scheme. A collection of numerical examples for linear equations are presented to verify our observations above. We also provide numerical examples based on non-uniform mesh, nonlinear Burgers’ equation, and high-dimensional Maxwell equations to explore superconvergence properties of DG methods in a more general setting.

Introduction

In this paper, we investigate superconvergence properties of discontinuous Galerkin (DG) and local DG (LDG) methods for smooth solutions of linear hyperbolic and parabolic problems. DG and LDG methods are a class of finite element methods, designed for solving hyperbolic and parabolic problems among many others [10]. These methods use piecewise polynomial spaces of degree k that could be discontinuous across cell boundaries as solution and test function spaces. These methods have the advantage of being compact and flexible for unstructured meshes, and being suitable for h-p adaptivity. Moreover, it has been proved that both DG and LDG methods are of order $k+1$ for linear problems with smooth enough solutions [10] for 1-D cases. For general meshes, DG solutions are proved to be $k+\frac{1}{2}$ order accurate for linear hyperbolic problems [18]. These methods also have some inherit dissipation mechanism for $L^2$ stability of nonlinear problems, see for example [14], [22] and references there in.

In this paper, our focus is on the superconvergence properties of DG and LDG solutions. Superconvergence properties of DG and LDG methods for hyperbolic and parabolic problems have been intensively investigated in the past. Lowrie et al. [17] discovered that when polynomials of degree k is used, “a component of error” of the DG method converges with order $2k+1$ in $L^2$ norm. It is showed in [9], [16] that the DG and LDG solutions converge with order $2k+1$ in terms of negative norm. Based on the negative norm estimate, the DG and LDG solutions on translation invariant grids can be post-processed via a kernel convolution with B-spline functions. The post-processed solution is proved to converge with order $2k+1$ in $L^2$ norm [9], [19], [16]. Adjerid et al. [1] analyzed the DG method in the setting of ordinary differential equations with the conclusion that the DG solution converges with order $k+2$ at Radau points of each element, and with order $2k+1$ at downwind points. In [2], Adjerid et al. numerically investigated the superconvergence of DG and LDG for convection–diffusion equations at Radau points. Cheng and Shu [5], [6], [7] showed that the DG and LDG solution is closer to the Radau projection of the exact solution than the exact solution itself. As a result, the error of DG and LDG solution will not grow over a long time period $\mathcal{O}\left(\frac{1}{\sqrt{\mathrm{\Delta }x}}\right)$. In [13], [3], [4], [20], [21], Fourier analysis has been adopted to indicate the superconvergence properties of DG solution in terms of dispersion and dissipation error of physically relevant eigenvalues. We remark that the work in [13], [3], [4] is based on initial-boundary value problem with a given inflow boundary condition, while our analysis in this paper is based on the initial-boundary value problem with a periodic boundary condition. Such difference leads to different assumptions, and therefore different observations and conclusions in the Fourier analysis. Zhong and Shu [26] use the Fourier analysis and symbolic computation to show that the DG method is superconvergent at Radau and downwind points with the order of $k+2$ and $2k+1$ respectively. In [23], for the first time that DG solutions are proved to converge at the optimal rate of $k+2$ at Radau points under the general assumption of non-uniform mesh. Because of these superconvergence properties, the method is considered to be very competitive in resolving waves propagating with long time integrations.

Different approaches have been adopted to analyze the superconvergence properties of DG schemes, such as the negative norm estimate [9], [16], by considering the problem as an initial or boundary value problem [1], [13], [3], [4], by special decomposition of error and playing with test functions in the weak formulation [5], [23], Fourier analysis [26], [20], [21], [12] etc. Fourier analysis has been known to be limited to linear problems with periodic boundary conditions and uniform mesh. However, it provides a sufficient condition for instability of “bad” schemes [25] as well as a quantitative error estimate. It can be used as a guidance to results in a more general setting [26]. In this paper, we will continue adopting Fourier approach to analyze the errors of DG and LDG methods for time dependent linear hyperbolic and parabolic equations.

In this paper, we perform Fourier analysis and symbolically compute eigenvalues and the corresponding eigenvectors of the amplification matrices of the DG and LDG methods with $P^k$ polynomial spaces. We obtain the following observations when $k=1\text{,}2\text{,}3$:

• There are $k+1$ eigenvalues of the amplification matrices for DG and LDG schemes. One of these eigenvalues is physically relevant. It approximates the analytical wave propagation speed with an order of $2k+1$ in dissipation error and an order of $2k+2$ in dispersion error for DG solutions; and with an order of $2k+2$ in dissipation error for LDG solutions. This is consistent with the results in [13], [3] for initial boundary value problems. The rest of the eigenvalues are non-physically relevant; they have large negative real part that is of order $\frac{1}{\mathrm{\Delta }x}$ for DG and $\frac{1}{\mathrm{\Delta }{x}^2}$ for LDG. As a result, the corresponding non-physically relevant eigenvectors will be damped exponentially fast with respect to $\mathrm{\Delta }x$ over time.

• There are $k+1$ eigenvectors. If Lagrangian basis functions based on shifted Radau points are used, the eigenvector corresponding to the physically relevant eigenvalue approximates the wave function with order $k+2$ at shifted Radau points and with order $2k+1$ at downwind points.

Based on the observations above, we decompose the error of DG and LDG solutions into three parts. One part is due to the dissipation and dispersion errors of physically relevant eigenvalues; this part of error is of very high order ($2k+1$ for DG and $2k+2$ for LDG) and will grow linearly in time. The second part is the projection error from the eigenvector analysis. The magnitude of this part of error does not grow in time. The third part of error is related to how non-physically relevant eigenvectors are being dissipated over time. It is concluded in this paper that the error of the DG or LDG solution at Radau points will not grow over a period of time that is on the order of $\frac{1}{\mathrm{\Delta }{x}^{k-1}}$ for DG solutions and is on the order of $\frac{1}{\mathrm{\Delta }{x}^k}$ for LDG solutions, where k is the degree of polynomial space. We remark that the numerical solution is closer to the special projection investigated in this paper than that in [5].

The paper is organized as follows. In Section 2, a review of DG and LDG methods and Fourier approach is given. In Section 3, we symbolically analyze the eigenstructure of the amplification matrices of the DG (Section 3.1) and LDG (Section 3.2) schemes with polynomial degrees up to k ($1⩽k⩽3$). We also comment on the supraconvergence of DG scheme based on Radau points in Section 3.3. We analyze the fully discrete RKDG scheme in Section 3.4. In Section 4, numerical examples for scalar and system of equations in one and two spatial dimensions are provided to verify our theoretical results. Numerical examples on equations with variable coefficients, nonlinear equations, as well as schemes based on non-uniform meshes are also presented to assess superconvergence properties of DG and LDG schemes in a more general setting. Some interesting observations are discussed based on our understanding. Conclusions are given in Section 5.