Discontinuous Galerkin Spatial Discretization Research Articles

Hp discontinuous Galerkin methods for the vertical extent of the water column in coastal settings part I: Barotropic forcing

In this article, we present novel, high-order, discontinuous Galerkin (DG) methods for the vertical extent of the water column in coastal settings. We examine the shallow water equations (SWE) in the context of DG spatial discretizations coupled with explicit Runge-Kutta (RK) time stepping. All the primary variables, including the free surface elevation, are discretized using discontinuous polynomial spaces of arbitrary order. The difficulty of mismatched lateral boundary faces that accompanies the use of a discontinuous free surface is overcome through the use of a so-called sigma-coordinate system in the vertical, which transforms the bottom boundary and free surface into coordinate surfaces. We develop high-order methods for the SWE that exhibit optimal orders of convergence for all the primary variables via two distinct paths: the first involves the use of a convolution kernel made up of B-splines to filter out errors in the DG discretization of the surface elevation and the corresponding pressure flux. The second involves a method that evaluates the discrete depth-integrated velocity exactly, eliminating the need to solve the depth-integrated momentum equation altogether. The result is a simple and efficient high-order scheme that can be extended to the full three-dimensional SWE.

A semi‐implicit, semi‐Lagrangian discontinuous Galerkin framework for adaptive numerical weather prediction

We present an adaptive discretization approach for model equations typical of numerical weather prediction (NWP), which combines the semi‐Lagrangian technique with a semi‐implicit time discretization method, based on the Trapezoidal Rule second‐order Backward Difference Formula scheme (TR‐BDF2), and with a discontinuous Galerkin (DG) spatial discretization, with variable and adaptive element degree. The resulting method has full second‐order accuracy in time, can employ polynomial bases of arbitrarily high degree in space, is unconditionally stable and can effectively adapt the number of degrees of freedom employed in each element at runtime, in order to balance accuracy and computational cost. Furthermore, although the proposed method can be implemented on arbitrary unstructured and non‐conforming meshes, even its application on simple Cartesian meshes in spherical coordinates can reduce the impact of the coordinate singularity, by reducing the polynomial degree used in the polar elements. Numerical results are presented, obtained on classical benchmarks with two‐dimensional models implementing discretizations of the shallow‐water equations on the sphere and of the Euler equations on a vertical slice, respectively. The results confirm that the proposed method has a significant potential for NWP applications.

Numerical study of the two-species Vlasov–Ampère system: Energy-conserving schemes and the current-driven ion-acoustic instability

In this paper, we propose energy-conserving Eulerian solvers for the two-species Vlasov–Ampère (VA) system and apply the methods to simulate current-driven ion-acoustic instability. The two-species VA systems are of practical importance in applications, and they conserve many physical quantities including the particle number of each species and the total energy that is comprised of kinetic energy for both species and the electric energy.The main goal of this paper is to generalize our previous work for the single-species VA system [9] and Vlasov–Maxwell (VM) system [8] to the two-species case. The methodologies proposed involve careful design of temporal discretization and the use of the discontinuous Galerkin (DG) spatial discretizations. We show that the energy-conserving time discretizations for single-species equations [9,8] can also work for the two-species case if extended properly. Compared to other high order schemes, we emphasize that our schemes can preserve the total particle number and total energy on the fully discrete level regardless of mesh size, making them very attractive for long time simulations. We benchmark our algorithms on a test example to check the one-species limit, and the current-driven ion-acoustic instability. To simulate the current-driven ion-acoustic instability, a slight modification for the implicit method is necessary to fully decouple the split equations. This is achieved by a Gauss–Seidel type iteration technique. Numerical results verified the conservation and performance of our methods. Finally, we remark that the schemes in this paper can be readily extended to applications when the models take more general form, such as the multi-species VM equations.

A weighted Runge–Kutta discontinuous Galerkin method for wavefield modelling

Summary In this paper, we develop a weighted Runge-Kutta discontinuous Galerkin method for the wavefield modeling, which is simply called the WRKDG method. For this method, we first transform the seismic wave equations into a first-order hyperbolic system, and then combine the discontinuous Galerkin spatial discretization with a weighted Runge-Kutta (RK) time discretization. The time discretization is based on a diagonal implicit RK method. We employ an explicit technique to change the implicit RK into an explicit method. We also investigate the stability criteria and numerical dispersion for solving the 2D acoustic equation for this method. Afterwards, we apply the WRKDG method to simulate the wavefields for a 2D homogeneous transversely isotropic (TI) model and a water-filled fracture model. Linear and quadratic interpolations are typically used in this article.

Fast Solver for the Local Discontinuous Galerkin Discretization of the KdV Type Equations

AbstractIn this paper, we will develop a fast iterative solver for the system of linear equations arising from the local discontinuous Galerkin (LDG) spatial discretization and additive Runge-Kutta (ARK) time marching method for the KdV type equations. Being implicit in time, the severe time step , with the k-th order of the partial differential equations (PDEs)) restriction for explicit methods will be removed. The equations at the implicit time level are linear and we demonstrate an efficient, practical multigrid (MG) method for solving the equations. In particular, we numerically show the optimal or sub-optimal complexity of the MG solver and a two-level local mode analysis is used to analyze the convergence behavior of the MG method. Numerical results for one-dimensional, two-dimensional and three-dimensional cases are given to illustrate the efficiency and capability of the LDG method coupled with the multigrid method for solving the KdV type equations.

High order asymptotic preserving DG-IMEX schemes for discrete-velocity kinetic equations in a diffusive scaling

In this paper, we develop a family of high order asymptotic preserving schemes for some discrete-velocity kinetic equations under a diffusive scaling, that in the asymptotic limit lead to macroscopic models such as the heat equation, the porous media equation, the advection–diffusion equation, and the viscous Burgers' equation. Our approach is based on the micro–macro reformulation of the kinetic equation which involves a natural decomposition of the equation to the equilibrium and non-equilibrium parts. To achieve high order accuracy and uniform stability as well as to capture the correct asymptotic limit, two new ingredients are employed in the proposed methods: discontinuous Galerkin (DG) spatial discretization of arbitrary order of accuracy with suitable numerical fluxes; high order globally stiffly accurate implicit–explicit (IMEX) Runge–Kutta scheme in time equipped with a properly chosen implicit–explicit strategy. Formal asymptotic analysis shows that the proposed scheme in the limit of ε→0 is a consistent high order discretization for the limiting equation. Numerical results are presented to demonstrate the stability and high order accuracy of the proposed schemes together with their performance in the limit. Our methods are also tested for the continuous-velocity one-group transport equation in slab geometry and for several examples with spatially varying parameters.

Relaxing the CFL Number of the Discontinuous Galerkin Method

We propose a family of high order methods for the solution of hyperbolic conservation laws which are based on the discontinuous Galerkin (DG) spatial discretization. In the standard DG method, the dispersion and dissipation errors and the spectrum of the semidiscrete scheme are related to the $[\frac{p}{p+1}]$ Pade approximants of $\exp(z)$ and $\exp(-z)$. These Pade approximants are responsible for the superconvergent $\mathcal{O}(h^{2p+2})$ and $\mathcal{O}(h^{2p+1})$ errors in dispersion and dissipation, respectively, and the restriction of the CFL number when increasing the order of approximation, $p$. By modifying the DG method we obtain different rational approximations of the exponential, thereby sacrificing some of the superconvergence of the method, and construct new schemes which allow larger time steps than the original DG method, while having the same order of convergence in the $\mathcal{L}^2$ norm. This is achieved through modifications to the numerical flux. The schemes preserve the attract...

Analysis of Asymptotic Preserving DG-IMEX Schemes for Linear Kinetic Transport Equations in a Diffusive Scaling

In this paper, some theoretical aspects will be addressed for the asymptotic preserving discontinuous Galerkin implicit-explicit (DG-IMEX) schemes recently proposed in [J. Jang, F. Li, J.-M. Qiu, and T. Xiong, High order asymptotic preserving DG-IMEX schemes for discrete-velocity kinetic equations in a diffusive scaling, http://arxiv.org/abs/1306.0227, 2013, submitted] for kinetic transport equations under a diffusive scaling. We will focus on the methods that are based on discontinuous Galerkin (DG) spatial discretizations with the $P^k$ polynomial space and a first order implicit-explicit (IMEX) temporal discretization, and apply them to two linear models: the telegraph equation and the one-group transport equation in slab geometry. In particular, we will establish uniform numerical stability with respect to Knudsen number $\varepsilon$ using energy methods, as well as error estimates for any given $\varepsilon$. When $\varepsilon\rightarrow 0$, a rigorous asymptotic analysis of the schemes is also obtained. Though the methods and the analysis are presented for one dimension in space, they can be generalized to higher dimensions directly.

Weighted Interior Penalty Method with Semi-Implicit Integration Factor Method for Non-Equilibrium Radiation Diffusion Equation

AbstractWeighted interior penalty discontinuous Galerkin method is developed to solve the two-dimensional non-equilibrium radiation diffusion equation on unstructured mesh. There are three weights including the arithmetic, the harmonic, and the geometric weight in the weighted discontinuous Galerkin scheme. For the time discretization, we treat the nonlinear diffusion coefficients explicitly, and apply the semi-implicit integration factor method to the nonlinear ordinary differential equations arising from discontinuous Galerkin spatial discretization. The semi-implicit integration factor method can not only avoid severe timestep limits, but also takes advantage of the local property of DG methods by which small sized nonlinear algebraic systems are solved element by element with the exact Newton iteration method. Numerical results are presented to demonstrate the validity of discontinuous Galerkin method for high nonlinear and tightly coupled radiation diffusion equation.

Optimal Strong-Stability-Preserving Runge–Kutta Time Discretizations for Discontinuous Galerkin Methods

Discontinuous Galerkin (DG) spatial discretizations are often used in a method-of-lines approach with explicit strong-stability-preserving (SSP) Runge–Kutta (RK) time steppers for the numerical solution of hyperbolic conservation laws. The time steps that are employed in this type of approach must satisfy Courant–Friedrichs–Lewy stability constraints that are dependent on both the region of absolute stability and the SSP coefficient of the RK method. While existing SSPRK methods have been optimized with respect to the latter, it is in fact the former that gives rise to stricter constraints on the time step in the case of RKDG stability. Therefore, in this work, we present the development of new “DG-optimized” SSPRK methods with stability regions that have been specifically designed to maximize the stable time step size for RKDG methods of a given order in one space dimension. These new methods represent the best available RKDG methods in terms of computational efficiency, with significant improvements over methods using existing SSPRK time steppers that have been optimized with respect to SSP coefficients. Second-, third-, and fourth-order methods with up to eight stages are presented, and their stability properties are verified through application to numerical test cases.

Locally Implicit Time Integration Strategies in a Discontinuous Galerkin Method for Maxwell’s Equations

An attractive feature of discontinuous Galerkin (DG) spatial discretization is the possibility of using locally refined space grids to handle geometrical details. However, locally refined meshes lead to severe stability constraints on explicit integration methods to numerically solve a time-dependent partial differential equation. If the region of refinement is small relative to the computational domain, the time step size restriction can be overcome by blending an implicit and an explicit scheme where only the solution variables living at fine elements are treated implicitly. The downside of this approach is having to solve a linear system per time step. But due to the assumed small region of refinement relative to the computational domain, the overhead will also be small while the solution can be advanced in time with step sizes determined by the coarse elements. In this paper, we present two locally implicit time integration methods for solving the time-domain Maxwell equations spatially discretized with a DG method. Numerical experiments for two-dimensional problems illustrate the theory and the usefulness of the implicit–explicit approaches in presence of local refinements.

Solution of Multimaterial Equilibrium Radiation Diffusion Problems by using the Discontinuous Galerkin Method

The discontinuous Galerkin method is used for solving the two-dimensional equilibrium radiation diffusion equation. We construct the weighted interior penalty method based on the geometric average weight. The semi-implicit integration factor method is applied to the nonlinear ordinary differential equations obtained by the discontinuous Galerkin spatial discretization. Numerical results are presented to demonstrate the validity and reliability of using the discontinuous Galerkin method for solving the highly nonlinear radiation diffusion equation.

An analysis of the spectrum of the discontinuous Galerkin method

We derive explicit expressions for the eigenvalues (spectrum) of the discontinuous Galerkin spatial discretization applied to the linear advection equation. We show that the eigenvalues are related to the subdiagonal [p/p+1] Padé approximation of exp(−z) when pth degree basis functions are used. We derive an upper bound on the eigenvalue with the largest magnitude as (p+1)(p+2). We demonstrate that this bound is not tight and prove that the asymptotic growth rate of the spectral radius is slower than quadratic in p. We also analyze the behavior of the spectrum near the imaginary axis to demonstrate that the spectral curves approach the imaginary axis although there are no purely imaginary eigenvalues.

Optimal Runge–Kutta schemes for discontinuous Galerkin space discretizations applied to wave propagation problems

We study the performance of methods of lines combining discontinuous Galerkin spatial discretizations and explicit Runge–Kutta time integrators, with the aim of deriving optimal Runge–Kutta schemes for wave propagation applications. We review relevant Runge–Kutta methods from literature, and consider schemes of order q from 3 to 4, and number of stages up to q + 4, for optimization. From a user point of view, the problem of the computational efficiency involves the choice of the best combination of mesh and numerical method; two scenarios are defined. In the first one, the element size is totally free, and a 8-stage, fourth-order Runge–Kutta scheme is found to minimize a cost measure depending on both accuracy and stability. In the second one, the elements are assumed to be constrained to such a small size by geometrical features of the computational domain, that accuracy is disregarded. We then derive one 7-stage, third-order scheme and one 8-stage, fourth-order scheme that maximize the stability limit. The performance of the three new schemes is thoroughly analyzed, and the benefits are illustrated with two examples. For each of these Runge–Kutta methods, we provide the coefficients for a 2 N-storage implementation, along with the information needed by the user to employ them optimally.

A Discontinuous Galerkin Method for Ideal Two-Fluid Plasma Equations

AbstractA discontinuous Galerkin method for the ideal 5 moment two-fluid plasma system is presented. The method uses a second or third order discontinuous Galerkin spatial discretization and a third order TVD Runge-Kutta time stepping scheme. The method is benchmarked against an analytic solution of a dispersive electron acoustic square pulse as well as the two-fluid electromagnetic shock [1] and existing numerical solutions to the GEM challenge magnetic reconnection problem [2]. The algorithm can be generalized to arbitrary geometries and three dimensions. An approach to maintaining small gauge errors based on error propagation is suggested.

Local-Structure-Preserving Discontinuous Galerkin Methods with Lax-Wendroff Type Time Discretizations for Hamilton-Jacobi Equations

In this paper, a family of high order numerical methods are designed to solve the Hamilton-Jacobi equation for the viscosity solution. In particular, the methods start with a hyperbolic conservation law system closely related to the Hamilton-Jacobi equation. The compact one-step one-stage Lax-Wendroff type time discretization is then applied together with the local-structure-preserving discontinuous Galerkin spatial discretization. The resulting methods have lower computational complexity and memory usage on both structured and unstructured meshes compared with some standard numerical methods, while they are capable of capturing the viscosity solutions of Hamilton-Jacobi equations accurately and reliably. A collection of numerical experiments is presented to illustrate the performance of the methods.

Semi discrete discontinuous Galerkin methods and stage-exceeding-order, strong-stability-preserving Runge–Kutta time discretizations

This paper investigates the use of a special class of strong-stability-preserving (SSP) Runge–Kutta time discretization methods in conjunction with discontinuous Galerkin (DG) finite element spatial discretizatons. The class of SSP methods investigated here is defined by the property that the number of stages s is greater than the order k of the method. From analysis, CFL conditions for the linear ( L 2) stability of the methods defined using the s > k SSP schemes are obtained that are less restrictive than those of the “standard” so-called RKDG methods that use s = k SSP Runge–Kutta schemes. The improvement in the CFL conditions for linear stability of the methods more than offsets the additional work introduced by the increased number of stages. Given that the CFL conditions for linear stability are what must be respected in practice in order to maintain high-order accuracy, the use of the s > k SSP schemes results in RKDG methods that are more efficient than those previously defined. Furthermore, with the application of a slope limiter, the nonlinear stability properties of the forward Euler method and the DG spatial discretization, which have been previously proven, are preserved with these methods under less restrictive CFL conditions than those required for linear stability. Thus, more efficient RKDG methods that possess the same favorable accuracy and stability properties of the “standard” RKDG methods are obtained. Numerical results verify the CFL conditions for stability obtained from analysis and demonstrate the efficiency advantages of these new RKDG methods.