Error Estimates for the Runge–Kutta Discontinuous Galerkin Method for the Transport Equation with Discontinuous Initial Data (original) (raw)
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Lecture# 1: Discontinuous Galerkin Methods: Motivation and Their Origin
In this lecture we introduce the main ideas of the discontinuous Galerkin approximations that are based on weak formulation of differential equation and the boundary conditions. First we introduce some notations and then we give a very simple example of one dimensional transport problem, which motivates and serves as a background for the approximation to the multidimensional transport problem. We discuss two equivalent weak formulations for the multidimensional problem. We conclude with some simple error estimates.
The presentation is concerned with the numerical treatment of transient transport problems like heat transfer or mass/species transport by means of discontinuous spatial discretization and different time integration schemes. To achieve a semidiscrete initial value problem discontinuous p-finite elements [1] for the approximation in space are used, where the continuity at the interelement boundaries is just weakly enforced. Furthermore, continuous and discontinuous Galerkin time integration schemes, which evaluate the balance equation in a weak sense over the time interval, are presented [7]. These discretization techniques are investigated with respect to robustness, reliability and accuracy, also in the context of non-smooth initial or boundary conditions. Selected benchmark analyses of heat conduction with an available analytical solution are analyzed for the above described numerical methods. Moreover the highly non-linear reaction-diffusion process of calcium leaching in cementi...
Journal of Computational Physics, 2006
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, nonoscillatory property, and resolution of discontinuities. Numerical tests are also performed for two dimensional systems.
Time step restrictions for Runge–Kutta discontinuous Galerkin methods on triangular grids
Journal of Computational Physics, 2008
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 ¼ 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.
The Runge–Kutta Discontinuous Galerkin Method for Conservation Laws V
Journal of Computational Physics, 1998
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.
An introduction to the discontinuous Galerkin method for convection-dominated problems
Advanced numerical approximation of nonlinear …, 1998
In these notes, we study the Runge Kutta Discontinuous Galerkin method for numericaly solving nonlinear hyperbolic systems and its extension for convectiondominated problems, the so-called Local Discontinuous Galerkin method. Examples of problems to which these methods can be applied are the Euler equations of gas dynamics, the shallow water equations, the equations of magneto-hydrodynamics, the compressible Navier-Stokes equations with high Reynolds numbers, and the equations of the hydrodynamic model for semiconductor device simulation. The main features that make the methods under consideration attractive are their formal highorder accuracy, their nonlinear stability, their high parallelizability, their ability to handle complicated geometries, and their ability to capture the discontinuities or strong gradients of the exact solution without producing spurious oscillations. The purpose of these notes is to provide a short introduction to the devising and analysis of these discontinuous Galerkin methods. Aknowledgements. The author is grateful to Al o Quarteroni for the invitation to give a series of lectures at the CIME, June 23{28, 1997, the material of which is contained in these notes. He also thanks F. Bassi and F. Rebay, and I. Lomtev and G.E. Karniadakis for kindly providing pictures from their papers 2] and 3], and 46] and 65], respectively. 1 2 Contents Preface Chapter 1. A historical overview 1.1. The original Discontinuous Galerkin method 1.2. Nonlinear hyperbolic systems: The RKDG method 1.3. Convection-di usion systems: The LDG method 1.4. The content of these notes Chapter 2. The scalar conservation law in one space dimension 2.1. Introduction 2.2. The discontinuous Galerkin-space discretization 2.3. The TVD-Runge-Kutta time discretization 2.4. The generalized slope limiter 2.5. Computational results 2.6. Concluding remarks 2.7. Appendix: Proof of the L 2-error estimates in the linear case Chapter 3. The RKDG method for multidimensional systems 3.
Journal of Computational Science, 2013
We revisit the finite element analysis of convection dominated flow problems within the recently developed Discontinuous Petrov-Galerkin (DPG) variational framework. We demonstrate how test function spaces that guarantee numerical stability can be computed automatically with respect to the so called optimal test space norm by using an element subgrid discretization. This should make the DPG method not only stable but also robust, that is, uniformly stable with respect to the Péclet number in the current application. The effectiveness of the algorithm is demonstrated on two problems for the linear advection-diffusion equation.
Discontinuous Galerkin methods for the solution of a class of hyperbolic problems
A numerical solution of one dimensional (1D) system of conservation laws is presented based on the Runge Kutta Discontinuous Galerkin (RKDG) method. We present and analyse the RKDG method and implement it for the Euler equations of gas dynamics and the shallow water equations for various initial data. The RKDG method is a very attractive method because of its formal high-order accuracy, its ability to handle complicated geometries, and its ability to capture discontinuities without producing spurious oscillations. Declaration I, the undersigned, hereby declare that the work contained in this research project is my original work, and that any work done by others or by myself previously has been acknowledged and referenced accordingly.
2012
This paper deals with a high-order accurate Runge Kutta Discontinuous Galerkin (RKDG) method for the numerical solution of the wave equation, which is one of the simple case of a linear hyperbolic partial differential equation. Nodal DG method is used for a finite element space discretization in 'x' by discontinuous approximations. This method combines mainly two key ideas which are based on the finite volume and finite element methods. The physics of wave propagation being accounted for by means of Riemann problems and accuracy is obtained by means of high-order polynomial approximations within the elements. High order accurate Low Storage Explicit Runge Kutta (LSERK) method is used for temporal discretization in 't' that allows the method to be nonlinearly stable regardless of its accuracy. The resulting RKDG methods are stable and high-order accurate. The L1 ,L2 and L∞ error norm analysis shows that the scheme is highly accurate and effective. Hence, the method is...