Kutta Discontinuous Galerkin Method Research Articles

Adaptive Runge–Kutta discontinuous Galerkin methods using different indicators: One-dimensional case

In this paper, we systematically investigate adaptive Runge–Kutta discontinuous Galerkin (RKDG) methods for hyperbolic conservation laws with different indicators which were based on the troubled cell indicators studied by Qiu and Shu [J. Qiu, C.-W. Shu, A comparison of troubled-cell indicators for Runge–Kutta discontinuous Galerkin mehtods using weighted essentially non-osillatory limiters, SIAM J. Sci. Comput. 27 (2005) 995–1013]. The emphasis is on comparison of the performance of adaptive RKDG method using different indicators, with an objective of obtaining efficient and reliable indicators to obtain better performance for adaptive computation to save computational cost. Both h-version and r-version adaptive methods are considered in the paper. The idea is to first identify “troubled cells” by different troubled-cell indicators, namely those cells where limiting might be needed and discontinuities might appear, then adopt an adaptive approach in these cells. A detailed numerical study in one-dimensional case is performed, addressing the issues of efficiency (less CPU cost and more accurate), non-oscillatory property, and resolution of discontinuities.

Well‐balancing issues related to the RKDG2 scheme for the shallow water equations

AbstractDiscontinuous Galerkin (DG) finite element methods have salient features that are mainly highlighted by their locality, their easiness in balancing the flux and source term gradients and their component‐wise structure. In the light of this, this paper aims to provide insights into the well‐balancing property of a second‐order Runge–Kutta Discontinuous Galerkin (RKDG2) method. For this purpose, a Godunov‐type RKDG2 method is presented for solving the shallow water equations. The scheme is based on local DG linear approximations and does not entail any special treatment of the source terms in order to achieve well‐balanced numerical results. The performance of the present RKDG2 scheme in reproducing conserved solutions for both free surface and discharge over strongly irregular topography is demonstrated by applying to several hydraulic benchmarks. Meanwhile, the effects of different slope limiting procedures on the well‐balancing property are investigated and discussed. This work may provide useful guidelines for developing a well‐balanced RKDG2 numerical scheme for shallow water flow simulation. Copyright © 2009 John Wiley & Sons, Ltd.

Dynamic p-adaptive Runge–Kutta discontinuous Galerkin methods for the shallow water equations

In this paper, dynamic p-adaptive Runge–Kutta discontinuous Galerkin (RKDG) methods for the two-dimensional shallow water equations (SWE) are investigated. The p-adaptive algorithm that is implemented dynamically adjusts the order of the elements of an unstructured triangular grid based on a simple measure of the local flow properties of the numerical solution. Time discretization is accomplished using optimal strong-stability-preserving (SSP) RK methods. The methods are tested on two idealized problems of coastal ocean modeling interest with complex bathymetry – namely, the idealization of a continental shelf break and a coastal inlet. Numerical results indicate the stability, robustness, and accuracy of the algorithm, and it is shown that the use of dynamic p-adaptive grids offers savings in CPU time relative to grids with elements of a fixed order p that use either local h-refinement or global p-refinement to adequately resolve the solution while offering comparable accuracy.

A confrontation of 1D and 2D RKDG numerical simulation of transitional flow at open‐channel junction

AbstractIn this study, a comparison between the 1D and 2D numerical simulation of transitional flow in open‐channel networks is presented and completely described allowing for a full comprehension of the modeling water flow. For flow in an open‐channel network, mutual effects exist among the channel branches joining at a junction. Therefore, for the 1D study, the whole system (branches and junction) cannot be treated individually. The 1D Saint Venant equations calculating the flow in the branches are then supplemented by various equations used at the junction: a discharge flow conservation equation between the branches arriving and leaving the junction, and a momentum or energy conservation equation. The disadvantages of the 1D study are that the equations used at the junction are of empirical nature due to certain parameters given by experimental results and moreover they often present a reduced field of validity. On the contrary, for the 2D study, the entire network is considered as a single unit and the flow in all the branches and junctions is solved simultaneously. Therefore, we simply apply the 2D Saint Venant equations, which are solved by a second‐order Runge–Kutta discontinuous Galerkin method. Finally, the experimental results obtained by Hager are used to validate and to compare the two approaches 1D and 2D. Copyright © 2009 John Wiley & Sons, Ltd.

Simulation of bipolar charge transport with trapping and recombination in polymeric insulators using Runge–Kutta discontinuous Galerkin method

The widely used QUICKEST method with ULTIMATE flux limiter is not capable of solving the charge transport problems with a very steep wavefront accurately, due to the wide stencil adopted. Furthermore, the splitting process of separating the convection and the reaction terms in the method introduces additional errors. To solve such problems accurately, a novel numerical method based on the Runge–Kutta discontinuous Galerkin (RKDG) method is introduced in this paper, which has high-order resolution and weak correlation between cells. The bipolar charge transport under dc voltage in solid dielectrics with trapping and recombination is simulated using this new method. The results of charge profiles provided by the method are obviously different from the simulation results in the existing literature. The method was verified by problems with analytical solution and experimental observations.

A discontinuous Galerkin method for the shallow water equations in spherical triangular coordinates

A global model of the atmosphere is presented governed by the shallow water equations and discretized by a Runge–Kutta discontinuous Galerkin method on an unstructured triangular grid. The shallow water equations on the sphere, a two-dimensional surface in R 3 , are locally represented in terms of spherical triangular coordinates, the appropriate local coordinate mappings on triangles. On every triangular grid element, this leads to a two-dimensional representation of tangential momentum and therefore only two discrete momentum equations. The discontinuous Galerkin method consists of an integral formulation which requires both area (elements) and line (element faces) integrals. Here, we use a Rusanov numerical flux to resolve the discontinuous fluxes at the element faces. A strong stability-preserving third-order Runge–Kutta method is applied for the time discretization. The polynomial space of order k on each curved triangle of the grid is characterized by a Lagrange basis and requires high-order quadature rules for the integration over elements and element faces. For the presented method no mass matrix inversion is necessary, except in a preprocessing step. The validation of the atmospheric model has been done considering standard tests from Williamson et al. [D.L. Williamson, J.B. Drake, J.J. Hack, R. Jakob, P.N. Swarztrauber, A standard test set for numerical approximations to the shallow water equations in spherical geometry, J. Comput. Phys. 102 (1992) 211–224], unsteady analytical solutions of the nonlinear shallow water equations and a barotropic instability caused by an initial perturbation of a jet stream. A convergence rate of O ( Δ x k + 1 ) was observed in the model experiments. Furthermore, a numerical experiment is presented, for which the third-order time-integration method limits the model error. Thus, the time step Δ t is restricted by both the CFL-condition and accuracy demands. Conservation of mass was shown up to machine precision and energy conservation converges for both increasing grid resolution and increasing polynomial order k.

Time step restrictions for Runge–Kutta discontinuous Galerkin methods on triangular grids

We derive CFL conditions for the linear stability of the so-called Runge–Kutta discontinuous Galerkin (RKDG) methods on triangular grids. Semidiscrete DG approximations using polynomials spaces of degree p = 0 , 1 , 2 , and 3 are considered and discretized in time using a number of different strong-stability-preserving (SSP) Runge–Kutta time discretization methods. Two structured triangular grid configurations are analyzed for wave propagation in different directions. Approximate relations between the two-dimensional CFL conditions derived here and previously established one-dimensional conditions can be observed after defining an appropriate triangular grid parameter h and a constant that is dependent on the polynomial degree p of the DG spatial approximation. Numerical results verify the CFL conditions that are obtained, and “optimal”, in terms of computational efficiency, two-dimensional RKDG methods of a given order are identified.

An improvement of classical slope limiters for high‐order discontinuous Galerkin method

AbstractIn this paper, we describe some existing slope limiters (Cockburn and Shu's slope limiter and Hoteit's slope limiter) for the two‐dimensional Runge–Kutta discontinuous Galerkin (RKDG) method on arbitrary unstructured triangular grids. We describe the strategies for detecting discontinuities and for limiting spurious oscillations near such discontinuities, when solving hyperbolic systems of conservation laws by high‐order discontinuous Galerkin methods. The disadvantage of these slope limiters is that they depend on a positive constant, which is, for specific hydraulic problems, difficult to estimate in order to eliminate oscillations near discontinuities without decreasing the high‐order accuracy of the scheme in the smooth regions. We introduce the idea of a simple modification of Cockburn and Shu's slope limiter to avoid the use of this constant number. This modification consists in: slopes are limited so that the solution at the integration points is in the range spanned by the neighboring solution averages. Numerical results are presented for a nonlinear system: the shallow water equations. Four hydraulic problems of discontinuous solutions of two‐dimensional shallow water are presented. The idealized dam break problem, the oblique hydraulic jump problem, flow in a channel with concave bed and the dam break problem in a converging–diverging channel are solved by using the different slope limiters. Numerical comparisons on unstructured meshes show a superior accuracy with the modified slope limiter. Moreover, it does not require the choice of any constant number for the limiter condition. Copyright © 2008 John Wiley & Sons, Ltd.

Simulation of supercritical flow in crossroads: Confrontation of a 2D and 3D numerical approaches to experimental results

This work deals with the modeling of a flood in an urban environment. Among the various types of urban flood events, it was decided to study specifically the severe surface flooding events, which take place in highly urbanized areas. This work concerns particularly the numerical resolution of the two-dimensional Saint Venant equations for the study of the propagation of flood through the crossroads in the city. A discontinuous finite-element space discretization with a second-order Runge–Kutta time discretization is used to solve the two-dimensional Saint Venant equations. The scheme is well suited to handle complicated geometries and requires a simple treatment of boundary conditions and source terms to obtain high-order accuracy. The explicit time integration, together with the use of orthogonal shape functions, makes the method for the investigated flows computationally more efficient than comparable second-order finite volume methods. The scheme is applied to several supercritical flows in crossroads, which are investigated by Mignot. The experimental results obtained by the author are used to verify the accuracy and the robustness of the method. The results obtained are compared to those obtained by a second-order finite volume method (Rubar20 (2D)) and by FLUENT (3D). A very good agreement between the numerical solution obtained by the Runge–Kutta discontinuous Galerkin (RKDG) method and the experimental measured data were found. The method is then able to simulate the flow patterns observed experimentally and able to predict well the water depths, the discharge distribution in the downstream branches of the crossroad and the location of the hydraulic jumps and other flow characteristics more than the other methods.

Riemann Solvers with Runge–Kutta Discontinuous Galerkin Schemes for the 1D Shallow Water Equations

The spectrum of this survey turns on the evaluation of some eminent Riemann solvers (or the so-called solver), for the shallow water equations, when employed with high-order Runge–Kutta discontinuous Galerkin (RKDG) methods. Based on the assumption that: The higher is the accuracy order of a numerical method, the less crucial is the choice of Riemann solver; actual literature rather use the Lax-Friedrich solver as it is easy and less costly, whereas many others could be also applied such as the Godunov, Roe, Osher, HLL, HLLC, and HLLE. In practical applications, the flow can be dominated by geometry, and friction effects have to be taken into consideration. With the intention of obtaining a suitable choice of the Riemann solver function for high-order RKDG methods, a one-dimensional numerical investigation was performed. Three traditional hydraulic problems were computed by this collection of solvers cooperated with high-order RKDG methods. A comparison of the performance of the solvers was carried out discussing the issue of L1-errors magnitude, CPU time cost, discontinuity resolution and source term effects.

Runge–Kutta discontinuous Galerkin method using WENO limiters II: Unstructured meshes

In [J. Qiu, C.-W. Shu, Runge–Kutta discontinuous Galerkin method using WENO limiters, SIAM Journal on Scientific Computing 26 (2005) 907–929], Qiu and Shu investigated using weighted essentially non-oscillatory (WENO) finite volume methodology as limiters for the Runge–Kutta discontinuous Galerkin (RKDG) methods for solving nonlinear hyperbolic conservation law systems on structured meshes. In this continuation paper, we extend the method to solve two-dimensional problems on unstructured meshes, with the goal of obtaining a robust and high order limiting procedure to simultaneously obtain uniform high order accuracy and sharp, nonoscillatory shock transition for RKDG methods. Numerical results are provided to illustrate the behavior of this procedure.

Error Estimates for the Runge–Kutta Discontinuous Galerkin Method for the Transport Equation with Discontinuous Initial Data

We study the approximation of nonsmooth solutions of the transport equation in one space dimension by approximations given by a Runge–Kutta discontinuous Galerkin method of order two. We take an initial datum, which has compact support and is smooth except at a discontinuity, and show that, if the ratio of the time step size to the grid size is less than $1/3$, the error at the time T in the $L^2(\mathbb{R}\setminus\mathcal{R}_T)$-norm is the optimal order two when $\mathcal{R}_T$ is a region of size $O(T^{1/2}\,h^{1/2}\;\log{1/h})$ to the right of the discontinuity and of size $O(T^{1/3}\,h^{2/3}\;\log{1/h})$ to the left. Numerical experiments validating these results are presented.

A well‐balanced Runge–Kutta discontinuous Galerkin method for the shallow‐water equations with flooding and drying

AbstractWe build and analyze a Runge–Kutta discontinuous Galerkin method to approximate the one‐ and two‐dimensional shallow‐water equations. We introduce a flux modification technique to derive a well‐balanced scheme preserving steady states at rest with variable ground elevation and a slope modification technique to deal satisfactorily with flooding and drying. Numerical results illustrating the performance of the proposed scheme are presented. Copyright © 2007 John Wiley & Sons, Ltd.

An adaptive high-order discontinuous Galerkin method with error control for the Hamilton–Jacobi equations. Part I: The one-dimensional steady state case

We propose and study an adaptive version of the discontinuous Galerkin method for Hamilton–Jacobi equations. It works as follows. Given the tolerance and the degree of the polynomial of the approximate solution, the adaptive algorithm finds a mesh on which the approximate solution has an L ∞-distance to the viscosity solution no bigger than the prescribed tolerance. The algorithm uses three main tools. The first is an iterative solver combining the explicit Runge–Kutta discontinuous Galerkin method and the implicit Newton’s method that enables us to solve the Hamilton–Jacobi equations efficiently. The second is a new a posteriori error estimate based on the approximate resolution of an approximate problem for the actual error. The third is a method that allows us to find a new mesh as a function of the old mesh and the ratio of the a posteriori error estimate to the tolerance. We display extensive numerical evidence that indicates that, for any given polynomial degree, the method achieves its goal with optimal complexity independently of the tolerance. This is done in the framework of one-dimensional steady-state model problems with periodic boundary conditions.

A Runge–Kutta discontinuous Galerkin method for viscous flow equations

This paper presents a Runge–Kutta discontinuous Galerkin (RKDG) method for viscous flow computation. The construction of the RKDG method is based on a gas-kinetic formulation, which not only couples the convective and dissipative terms together, but also includes both discontinuous and continuous representation in the flux evaluation at a cell interface through a simple hybrid gas distribution function. Due to the intrinsic connection between the gas-kinetic BGK model and the Navier–Stokes equations, the Navier–Stokes flux is automatically obtained by the present method. Numerical examples for both one dimensional (1D) and two dimensional (2D) compressible viscous flows are presented to demonstrate the accuracy and shock capturing capability of the current RKDG method.

A Runge–Kutta discontinuous Galerkin method for linear free-surface gravity waves using high order velocity recovery

We present a higher order accurate discontinuous Galerkin finite element method for the simulation of linear free-surface gravity waves. The method uses the classical Runge–Kutta method for the time-discretization of the free-surface equations and the discontinuous Galerkin method for the space-discretization. In order to circumvent numerical instabilities arising from an asymmetric mesh a stabilization term is added to the free-surface equations. In combination with a higher order velocity recovery technique this stabilizes the numerical discretization with minimal effect on the accuracy of the wave computations. A stability analysis of the semi and fully-discrete scheme is presented, which suggests that for a suitable choice of the stabilization constant a relatively large time step can be chosen for accurate simulations over a long period of time. Numerical examples of a number of problems are also presented.

Error Control for a Class of Runge–Kutta Discontinuous Galerkin Methods for Nonlinear Conservation Laws

We propose an a posteriori error estimate for the Runge-Kutta discontinuous Galerkin method (RK-DG) of arbitrary order in arbitrary space dimensions. For stabilization of the scheme a general framework of projections is introduced. Finally it is demonstrated numerically how the a posteriori error estimate is used to design both an efficient grid adaption and gradient limiting strategy. Numerical experiments show the stability of the scheme and the gain in efficiency in comparison with computations on uniform grids.

Runge–Kutta discontinuous Galerkin methods for compressible two-medium flow simulations: One-dimensional case

The Runge–Kutta discontinuous Galerkin (RKDG) method for solving hyperbolic conservation laws is a high order finite element method, which utilizes the useful features from high resolution finite volume schemes, such as the exact or approximate Riemann solvers, TVD Runge–Kutta time discretizations, and limiters. In this paper, we investigate using the RKDG finite element method for compressible two-medium flow simulation with conservative treatment of the moving material interfaces. Numerical results for both gas–gas and gas–water flows in one-dimension are provided to demonstrate the characteristic behavior of this approach.

Error Estimates to Smooth Solutions of Runge–Kutta Discontinuous Galerkin Method for Symmetrizable Systems of Conservation Laws

In this paper we study the error estimates to sufficiently smooth solutions of symmetrizable systems of conservation laws for the Runge–Kutta discontinuous Galerkin (RKDG) method. Time discretization is the second‐order explicit TVD (total variation diminishing) Runge–Kutta method, and the $\mathbb{P}^k$ (piecewise polynomial) finite element is used. When $k=1$ (piecewise linear finite element), the error estimate is obtained under the usual CFL condition $\dt\leq \beta h$ for nonlinear systems in one dimension and for linear systems in multiple space dimensions. Here, h is the maximum element length, τ is the time step, and β is a positive constant independent of h and τ. Error estimates for $\mathbb{P}^k$ finite elements with $k>1$ are obtained under a more restrictive CFL condition.

A numerical study for the performance of the Runge–Kutta discontinuous Galerkin method based on different numerical fluxes

Runge–Kutta discontinuous Galerkin (RKDG) method is a high order finite element method for solving hyperbolic conservation laws employing useful features from high resolution finite volume schemes, such as the exact or approximate Riemann solvers serving as numerical fluxes, TVD Runge–Kutta time discretizations, and limiters. In most of the RKDG papers in the literature, the Lax–Friedrichs numerical flux is used due to its simplicity, although there are many other numerical fluxes which could also be used. In this paper, we systematically investigate the performance of the RKDG method based on different numerical fluxes, including the first-order monotone fluxes such as the Godunov flux, the Engquist–Osher flux, etc., and second-order TVD fluxes, with the objective of obtaining better performance by choosing suitable numerical fluxes. The detailed numerical study is mainly performed for the one dimensional system case, addressing the issues of CPU cost, accuracy, non-oscillatory property, and resolution of discontinuities. Numerical tests are also performed for two dimensional systems.