The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case (original) (raw)
Related papers
Mathematics of Computation, 1989
This is the second paper in a series in which we construct and analyze a class of TVB (total variation bounded) discontinuous Galerkin finite element methods for solving conservation laws ut + J^ =1(fi(u))Xi = 0. In this paper we present a general framework of the methods, up to any order of formal accuracy, using scalar one-dimensional initial value and initial-boundary problems as models. In these cases we prove TVBM (total variation bounded in the means), TVB, and convergence of the schemes. Numerical results using these methods are also given. Extensions to systems and/or higher dimensions will appear in future papers.
Journal of Computational Physics, 1989
This is the third paper in a series in which we construct and analyze a class of TVB (total variation bounded) discontinuous Galerkin finite element methods for solving conservation laws II, + zy= ,(f,(u)), = 0. In this paper we present the method in a system of equations, stressing the point of how to use the weak form in the component spaces, but to use the local projection limiting in the characteristic fields, and how to implement boundary conditions. A l-dimensional system is thus chosen as a model. Different implementation techniques are discussed, theories analogous to scalar cases are proven for linear systems, and numerical results are given illustrating the method on nonlinear systems. Discussions of handling complicated geometries via adaptive triangle elements will appear in future papers. 0 1989 Academic Press. Inc.
1st National Fluid Dynamics Conference, 1988
L'accès aux archives de la revue « Modélisation mathématique et analyse numérique » implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/
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.
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.
A Problem-Independent Limiter for High-Order Runge–Kutta Discontinuous Galerkin Methods
2001
This paper is devoted to the use of discontinuous Galerkin methods to solve hyperbolic conservation laws. The emphasis is laid on the elaboration of slope limiters to enforce nonlinear stability for shock-capturing. The objectives are to derive problem-independent methods that maintain high-order of accuracy in regions where the solution is smooth, and in the neighborhood of shock waves. The aim is also to define a way of taking into account high-order space discretization in limiting process, to make use of all the expansion terms of the approximate solution. A new slope limiter is first presented for one-dimensional problems and any order of approximation. Next, it is extended to bidimensional problems, for unstructured triangular meshes. The new method is totally free of problem-dependence. Numerical experiments show its capacity to preserve the accuracy of discontinuous Galerkin method in smooth regions, and to capture strong shocks.
Application of implicit–explicit high order Runge–Kutta methods to discontinuous-Galerkin schemes
Journal of Computational Physics, 2007
Despite the popularity of high-order explicit Runge-Kutta (ERK) methods for integrating semi-discrete systems of equations, ERK methods suffer from severe stability-based time step restrictions for very stiff problems. We implement a discontinuous Galerkin finite element method (DGFEM) along with recently introduced high-order implicit-explicit Runge-Kutta (IMEX-RK) schemes to overcome geometry-induced stiffness in fluid-flow problems. The IMEX algorithms solve the non-stiff portions of the domain using explicit methods, and isolate and solve the more expensive stiff portions using an L-stable, stiffly-accurate explicit, singly diagonally implicit Runge-Kutta method (ESDIRK). Furthermore, we apply adaptive time-step controllers based on the embedded temporal error predictors. We demonstrate in a number of numerical test problems that IMEX methods in conjunction with efficient preconditioning become more efficient than explicit methods for systems exhibiting high levels of grid-induced stiffness.
Journal of Computational Physics, 2012
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, fourthorder 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 2N-storage implementation, along with the information needed by the user to employ them optimally.
Discontinuous Galerkin Methods for Computational Fluid Dynamics
2004
The discontinuous Galerkin methods are locally conservative, high-order accurate, and robust methods that can easily handle elements of arbitrary shapes, irregular triangulations with hanging nodes, and polynomial approximations of different degrees in different elements. These properties, which render them ideal for hp-adaptivity in domains of complex geometry, have brought them to the main stream of computational fluid dynamics. In this paper, we study the properties of the DG methods as applied to a wide variety of problems including linear, symmetric hyperbolic systems, the Euler equations of gas dynamics, purely elliptic problems and the incompressible and compressible Navier-Stokes equations. In each instance, we discuss the main properties of the methods, display the mechanisms that make them work so well, and present numerical experiments showing their performance.