Table of Content

20 March 2022, Volume 4 Issue 1

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PREFACE

Preface to Focused Issue on Discontinuous Galerkin Methods

Jan S. Hesthaven, Jennifer Ryan, Chi-Wang Shu, Jaap van der Vegt, Yan Xu, Qiang Zhang, Zhimin Zhang

2022, 4(1):  1-2.  doi:10.1007/s42967-021-00170-1

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Comparison of Semi-Lagrangian Discontinuous Galerkin Schemes for Linear and Nonlinear Transport Simulations

Xiaofeng Cai, Wei Guo, Jing-Mei Qiu

2022, 4(1):  3-33.  doi:10.1007/s42967-020-00088-0

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Transport problems arise across diverse felds of science and engineering. Semi-Lagrangian (SL) discontinuous Galerkin (DG) methods are a class of high-order deterministic transport solvers that enjoy advantages of both the SL approach and the DG spatial discretization. In this paper, we review existing SLDG methods to date and compare numerically their performance. In particular, we make a comparison between the splitting and nonsplitting SLDG methods for multi-dimensional transport simulations. Through extensive numerical results, we ofer a practical guide for choosing optimal SLDG solvers for linear and nonlinear transport simulations.

Conservative Discontinuous Galerkin/Hermite Spectral Method for the Vlasov-Poisson System

Francis Filbet, Tao Xiong

2022, 4(1):  34-59.  doi:10.1007/s42967-020-00089-z

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We propose a class of conservative discontinuous Galerkin methods for the Vlasov-Poisson system written as a hyperbolic system using Hermite polynomials in the velocity variable. These schemes are designed to be systematically as accurate as one wants with provable conservation of mass and possibly total energy. Such properties in general are hard to achieve within other numerical method frameworks for simulating the Vlasov-Poisson system. The proposed scheme employs the discontinuous Galerkin discretization for both the Vlasov and the Poisson equations, resulting in a consistent description of the distribution function and the electric feld. Numerical simulations are performed to verify the order of the accuracy and conservation properties.

An Adaptive Multiresolution Ultra-weak Discontinuous Galerkin Method for Nonlinear Schrödinger Equations

Zhanjing Tao, Juntao Huang, Yuan Liu, Wei Guo, Yingda Cheng

2022, 4(1):  60-83.  doi:10.1007/s42967-020-00096-0

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This paper develops a high-order adaptive scheme for solving nonlinear Schrödinger equations. The solutions to such equations often exhibit solitary wave and local structures, which make adaptivity essential in improving the simulation efciency. Our scheme uses the ultra-weak discontinuous Galerkin (DG) formulation and belongs to the framework of adaptive multiresolution schemes. Various numerical experiments are presented to demonstrate the excellent capability of capturing the soliton waves and the blow-up phenomenon.

A Local Discontinuous Galerkin Method with Generalized Alternating Fluxes for 2D Nonlinear Schrödinger Equations

Hongjuan Zhang, Boying Wu, Xiong Meng

2022, 4(1):  84-107.  doi:10.1007/s42967-020-00100-7

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In this paper, we consider the local discontinuous Galerkin method with generalized alternating numerical fuxes for two-dimensional nonlinear Schrödinger equations on Cartesian meshes. The generalized fuxes not only lead to a smaller magnitude of the errors, but can guarantee an energy conservative property that is useful for long time simulations in resolving waves. By virtue of generalized skew-symmetry property of the discontinuous Galerkin spatial operators, two energy equations are established and stability results containing energy conservation of the prime variable as well as auxiliary variables are shown. To derive optimal error estimates for nonlinear Schrödinger equations, an additional energy equation is constructed and two a priori error assumptions are used. This, together with properties of some generalized Gauss-Radau projections and a suitable numerical initial condition, implies optimal order of k + 1. Numerical experiments are given to demonstrate the theoretical results.

A Wavelet-Free Approach for Multiresolution-Based Grid Adaptation for Conservation Laws

Nils Gerhard, Siegfried Müller, Aleksey Sikstel

2022, 4(1):  108-142.  doi:10.1007/s42967-020-00101-6

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In recent years the concept of multiresolution-based adaptive discontinuous Galerkin (DG) schemes for hyperbolic conservation laws has been developed. The key idea is to perform a multiresolution analysis of the DG solution using multiwavelets defned on a hierarchy of nested grids. Typically this concept is applied to dyadic grid hierarchies where the explicit construction of the multiwavelets has to be performed only for one reference element. For non-uniform grid hierarchies multiwavelets have to be constructed for each element and, thus, becomes extremely expensive. To overcome this problem a multiresolution analysis is developed that avoids the explicit construction of multiwavelets.

Goal-Oriented Anisotropic hp-Adaptive Discontinuous Galerkin Method for the Euler Equations

Vít Dolejší, Filip Roskovec

2022, 4(1):  143-179.  doi:10.1007/s42967-020-00102-5

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We deal with the numerical solution of the compressible Euler equations with the aid of the discontinuous Galerkin (DG) method with focus on the goal-oriented error estimates and adaptivity. We analyse the adjoint consistency of the DG scheme where the adjoint problem is not formulated by the diferentiation of the DG form and the target functional but using a suitable linearization of the nonlinear forms. Furthermore, we present the goaloriented anisotropic hp-mesh adaptation method for the Euler equations. The theoretical results are supported by numerical experiments.

Superconvergence Study of the Direct Discontinuous Galerkin Method and Its Variations for Difusion Equations

Yuqing Miao, Jue Yan, Xinghui Zhong

2022, 4(1):  180-204.  doi:10.1007/s42967-020-00107-0

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In this paper, we apply the Fourier analysis technique to investigate superconvergence properties of the direct disontinuous Galerkin (DDG) method (Liu and Yan in SIAM J Numer Anal 47(1):475-698, 2009), the DDG method with the interface correction (DDGIC) (Liu and Yan in Commun Comput Phys 8(3):541-564, 2010), the symmetric DDG method (Vidden and Yan in Comput Math 31(6):638-662, 2013), and the nonsymmetric DDG method (Yan in J Sci Comput 54(2):663-683, 2013). We also include the study of the interior penalty DG (IPDG) method, due to its close relation to DDG methods. Error estimates are carried out for both P² and P³ polynomial approximations. By investigating the quantitative errors at the Lobatto points, we show that the DDGIC and symmetric DDG methods are superior, in the sense of obtaining (k + 2)th superconvergence orders for both P² and P³ approximations. Superconvergence order of (k + 2) is also observed for the IPDG method with P³ polynomial approximations. The errors are sensitive to the choice of the numerical fux coefcient for even degree P² approximations, but are not for odd degree P³ approximations. Numerical experiments are carried out at the same time and the numerical errors match well with the analytically estimated errors.

A Uniformly Robust Staggered DG Method for the Unsteady Darcy-Forchheimer-Brinkman Problem

Lina Zhao, Ming Fai Lam, Eric Chung

2022, 4(1):  205-226.  doi:10.1007/s42967-020-00106-1

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In this paper, we propose and analyze a uniformly robust staggered DG method for the unsteady Darcy-Forchheimer-Brinkman problem. Our formulation is based on velocity gradient-velocity-pressure and the resulting scheme can be fexibly applied to fairly general polygonal meshes. We relax the tangential continuity for velocity, which is the key ingredient in achieving the uniform robustness. We present well-posedness and error analysis for both the semi-discrete scheme and the fully discrete scheme, and the theories indicate that the error estimates for velocity are independent of pressure. Several numerical experiments are presented to confrm the theoretical fndings.

Local Discontinuous Galerkin Methods with Novel Basis for Fractional Difusion Equations with Non-smooth Solutions

Liyao Lyu, Zheng Chen

2022, 4(1):  227-249.  doi:10.1007/s42967-020-00104-3

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In this paper, we develop novel local discontinuous Galerkin (LDG) methods for fractional difusion equations with non-smooth solutions. We consider such problems, for which the solutions are not smooth at boundary, and therefore the traditional LDG methods with piecewise polynomial solutions sufer accuracy degeneracy. The novel LDG methods utilize a solution information enriched basis, simulate the problem on a paired special mesh, and achieve optimal order of accuracy. We analyze the L² stability and optimal error estimate in L²-norm. Finally, numerical examples are presented for validating the theoretical conclusions.

Negative Norm Estimates for Arbitrary Lagrangian-Eulerian Discontinuous Galerkin Method for Nonlinear Hyperbolic Equations

Qi Tao, Yan Xu, Xiaozhou Li

2022, 4(1):  250-270.  doi:10.1007/s42967-020-00108-z

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In this paper, we present the negative norm estimates for the arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) method solving nonlinear hyperbolic equations with smooth solutions. The smoothness-increasing accuracy-conserving (SIAC) flter is a postprocessing technique to enhance the accuracy of the discontinuous Galerkin (DG) solutions. This work is the essential step to extend the SIAC flter to the moving mesh for nonlinear problems. By the post-processing theory, the negative norm estimates are vital to get the superconvergence error estimates of the solutions after post-processing in the L² norm. Although the SIAC flter has been extended to nonuniform mesh, the analysis of fltered solutions on the nonuniform mesh is complicated. We prove superconvergence error estimates in the negative norm for the ALE-DG method on moving meshes. The main diffculties of the analysis are the terms in the ALE-DG scheme brought by the grid velocity feld, and the time-dependent function space. The mapping from time-dependent cells to reference cells is very crucial in the proof. The numerical results also confrm the theoretical proof.

The Direct Discontinuous Galerkin Methods with Implicit-Explicit Runge-Kutta Time Marching for Linear Convection-Difusion Problems

Haijin Wang, Qiang Zhang

2022, 4(1):  271-292.  doi:10.1007/s42967-020-00114-1

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In this paper, a fully discrete stability analysis is carried out for the direct discontinuous Galerkin (DDG) methods coupled with Runge-Kutta-type implicit-explicit time marching, for solving one-dimensional linear convection-difusion problems. In the spatial discretization, both the original DDG methods and the refned DDG methods with interface corrections are considered. In the time discretization, the convection term is treated explicitly and the difusion term implicitly. By the energy method, we show that the corresponding fully discrete schemes are unconditionally stable, in the sense that the time-step τ is only required to be upper bounded by a constant which is independent of the mesh size h. Optimal error estimate is also obtained by the aid of a special global projection. Numerical experiments are given to verify the stability and accuracy of the proposed schemes.

A Compatible Embedded-Hybridized Discontinuous Galerkin Method for the Stokes-Darcy-Transport Problem

Aycil Cesmelioglu, Sander Rhebergen

2022, 4(1):  293-318.  doi:10.1007/s42967-020-00115-0

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We present a stability and error analysis of an embedded-hybridized discontinuous Galerkin (EDG-HDG) finite element method for coupled Stokes-Darcy flow and transport. The flow problem, governed by the Stokes-Darcy equations, is discretized by a recently introduced exactly mass conserving EDG-HDG method while an embedded discontinuous Galerkin (EDG) method is used to discretize the transport equation. We show that the coupled flow and transport discretization are compatible and stable. Furthermore, we show the existence and uniqueness of the semi-discrete transport problem and develop optimal a priori error estimates. We provide numerical examples illustrating the theoretical results. In particular, we compare the compatible EDG-HDG discretization to a discretization of the coupled Stokes-Darcy and transport problem that is not compatible. We demonstrate that where the incompatible discretization may result in spurious oscillations in the solution to the transport problem, the compatible discretization is free of oscillations. An additional numerical example with realistic parameters is also presented.

Superconvergence Analysis of the Runge-Kutta Discontinuous Galerkin Method with Upwind-Biased Numerical Flux for Two-Dimensional Linear Hyperbolic Equation

Yuan Xu, Qiang Zhang

2022, 4(1):  319-352.  doi:10.1007/s42967-020-00116-z

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In this paper, we shall establish the superconvergence properties of the Runge-Kutta discontinuous Galerkin method for solving two-dimensional linear constant hyperbolic equation, where the upwind-biased numerical fux is used. By suitably defning the correction function and deeply understanding the mechanisms when the spatial derivatives and the correction manipulations are carried out along the same or diferent directions, we obtain the superconvergence results on the node averages, the numerical fuxes, the cell averages, the solution and the spatial derivatives. The superconvergence properties in space are preserved as the semi-discrete method, and time discretization solely produces an optimal order error in time. Some numerical experiments also are given.

Maximum-Principle-Preserving Local Discontinuous Galerkin Methods for Allen-Cahn Equations

Jie Du, Eric Chung, Yang Yang

2022, 4(1):  353-379.  doi:10.1007/s42967-020-00118-x

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In this paper, we study the classical Allen-Cahn equations and investigate the maximumprinciple-preserving (MPP) techniques. The Allen-Cahn equation has been widely used in mathematical models for problems in materials science and fuid dynamics. It enjoys the energy stability and the maximum-principle. Moreover, it is well known that the AllenCahn equation may yield thin interface layer, and nonuniform meshes might be useful in the numerical solutions. Therefore, we apply the local discontinuous Galerkin (LDG) method due to its fexibility on h-p adaptivity and complex geometry. However, the MPP LDG methods require slope limiters, then the energy stability may not be easy to obtain. In this paper, we only discuss the MPP technique and use numerical experiments to demonstrate the energy decay property. Moreover, due to the stif source given in the equation, we use the conservative modifed exponential Runge-Kutta methods and thus can use relatively large time step sizes. Thanks to the conservative time integration, the bounds of the unknown function will not decay. Numerical experiments will be given to demonstrate the good performance of the MPP LDG scheme.