Superconvergence of discontinuous Galerkin and local discontinuous Galerkin methods: Eigen-structure analysis based on Fourier approach
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 for linear problems with smooth enough solutions [10] for 1-D cases. For general meshes, DG solutions are proved to be order accurate for linear hyperbolic problems [18]. These methods also have some inherit dissipation mechanism for 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 in norm. It is showed in [9], [16] that the DG and LDG solutions converge with order 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 in 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 at Radau points of each element, and with order 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 . 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 and respectively. In [23], for the first time that DG solutions are proved to converge at the optimal rate of 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 polynomial spaces. We obtain the following observations when :
- 1.
There are 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 in dissipation error and an order of in dispersion error for DG solutions; and with an order of 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 for DG and for LDG. As a result, the corresponding non-physically relevant eigenvectors will be damped exponentially fast with respect to over time.
- 2.
There are 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 at shifted Radau points and with order at downwind points.
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 (). 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.
Section snippets
DG scheme
We first review the DG formulation and Fourier analysis for the linear hyperbolic problemwith periodic boundary conditions. Here a is a constant indicating wave propagation speed and is the wave number. For convenience, assume that .
To define the DG method for the model problem, we consider a uniform partition of the computational domain into N cells as follows:Denote the cell by and the cell center
1-D scalar and system of linear hyperbolic equations
Depending on different choices of basis functions in DG implementation, the amplification matrix G could be different. The eigenvalues of G however will stay the same, since the DG method is independent of the choice of basis functions. However, the eigenvectors will be basis-dependent. Below we analyze G matrix based on the basis functions that are the Lagrangian polynomialswhereare the shifted Radau points. are the roots of the Radau
Numerical examples
In this section, we provide a collection of one- and two-dimensional numerical experiments to verify our theoretical analysis in Section 3. DG schemes for a one-dimensional linear equation based on non-uniform mesh, one-dimensional nonlinear Burgers’ equation, two-dimensional systems such as wave equations and Maxwell equations are also investigated to explore superconvergence properties of DG methods in a more general setting. We do not report DG errors at Radau points due to superconvergence
Conclusion
In this paper, we discussed the superconvergence properties of discontinuous Galerkin (DG) and local DG (LDG) methods for linear hyperbolic and parabolic problems via Fourier approach. Especially, superconvergence properties of DG with uniform mesh for linear problems with periodic boundary conditions are discussed in terms of.
- •
dissipation and dispersion error of physically relevant eigenvalues; this part of error is related to the negative norm of DG error as discussed in [9], [16]
- •
eigenvectors
Acknowledgements
The authors thank Prof. Chi-Wang Shu for many helpful discussions and suggestions. The research is being conducted when W. Guo and J.-M. Qiu are visiting Institute of Computational and Experimental Research in Mathematics (ICERM) in Brown University. The support from ICERM is greatly appreciated. The research of the first and third author are supported by Air Force Office of Scientific Computing YIP grants FA9550-12-0318, NSF, DMS-0914852 and DMS-1217008, University of Houston.
References (26)
- et al.
A posteriori error estimation for discontinuous Galerkin solutions of hyperbolic problems
Computer Methods in Applied Mechanics and Engineering
(2002) - et al.
Superconvergence of discontinuous Galerkin solutions for a nonlinear scalar hyperbolic problem
Computer Methods in Applied Mechanics and Engineering
(2006) Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods
Journal of Computational Physics
(2004)- et al.
Superconvergence and time evolution of discontinuous Galerkin finite element solutions
Journal of Computational Physics
(2008) - et al.
Superconvergence of local discontinuous Galerkin methods for one-dimensional convection–diffusion equations
Computers and Structures
(2009) - et al.
Locally divergence-free discontinuous Galerkin methods for the Maxwell equations
Journal of Computational Physics
(2004) - et al.
An analysis of the discontinuous Galerkin method for wave propagation problems
Journal of Computational Physics
(1999) - et al.
An analysis of and a comparison between the discontinuous Galerkin and the spectral finite volume methods
Computers and fluids
(2005) - et al.
Numerical resolution of discontinuous Galerkin methods for time dependent wave equations
Computer Methods in Applied Mechanics and Engineering
(2011) - et al.
Dispersive and dissipative properties of discontinuous Galerkin finite element methods for the second-order wave equation
Journal of Scientific Computing
(2006)
Superconvergence of discontinuous Galerkin and local discontinuous Galerkin schemes for linear hyperbolic and convection–diffusion equations in one space dimension
SIAM Journal on Numerical Analysis
Enhanced accuracy by post-processing for finite element methods for hyperbolic equations
Mathematics of Computation
Runge–Kutta discontinuous Galerkin methods for convection-dominated problems
Journal of Scientific Computing
Cited by (55)
Superconvergence points for higher-order derivative interpolation and its applications in spectral collocation method
2024, Journal of Mathematical Analysis and ApplicationsEigensolution analysis of immersed boundary method based on volume penalization: Applications to high-order schemes
2022, Journal of Computational PhysicsSweep-Net: An Artificial Neural Network for radiation transport solves
2021, Journal of Computational PhysicsEnhancing accuracy with a convolution filter: What works and why!
2020, Computers and FluidsAnalysis of Recovery-assisted discontinuous Galerkin methods for the compressible Navier-Stokes equations
2020, Journal of Computational PhysicsCitation Excerpt :Fourier analysis is applied to examine the properties of the Recovery-assisted scheme and compare it to the conventional DG approach for advection-diffusion equations. Each scheme is expressed as the update matrix of a linear dynamical system whose properties reflect those of the numerical scheme itself (see Guo et al. [30] or Zhang & Shu [31] for clarification). Analysis of the update matrix yields three pieces of information: first, the spectral radius (which governs the maximum stable timestep size in explicit time integration), second, the resolving efficiency, and third, the order of accuracy.
Transformation to a block-diagonal form of matrices generating bounded semigroups
2020, Linear Algebra and Its Applications