Element Method For Conservation Laws Research Articles

A characteristic space–time conservation element and solution element method for conservation laws II. Multidimensional extension

The characteristic space-time conservation element and solution element (CE/SE) schemes proposed by Shen et al. (2015) 15 are straightforward extended to multidimensional schemes on 2D rectangular meshes which strictly follow the space-time conservation law. The schemes for solving both scalar conservation laws and compressible Euler equations with shock waves are developed. They are accurate and robust with CFL widely ranging from 0 to nearly 1. In Euler solvers, the frequently-used Harten, Lax and van Leer (HLL) Riemann solver, contact discontinuity restoring HLLC Riemann solver and Roe Riemann solver are employed to calculate the upwind fluxes as examples. When standard grid-aligned Riemann solvers are employed, the carbuncle phenomena are significantly suppressed when comparing with conventional upwind schemes. If rotated Riemann solvers are employed, nearly carbuncle-free results are obtained. Several well understood numerical examples are carried out to demonstrate that the 2D characteristic CE/SE schemes can simultaneously capture shocks and details of complex flow structures very well.

A characteristic space–time conservation element and solution element method for conservation laws

In this paper, an upwind space–time conservation element and solution element (CE/SE) method is developed to solve conservation laws. In the present method, the mesh quantity and spatial derivatives are the independent marching variables, which is consistent with the original CE/SE method proposed by Chang (1995) [5]. The staggered time marching strategy and the definition of conservation element (CE) also follow Chang's propositions. Nevertheless, the definition of solution element (SE) is modified from that of Chang. The numerical flux through the interface of two different conservation elements is not directly derived by a Taylor expansion in the reversed time direction as proposed by Chang, but determined by an upwind procedure. This modification does not change the local and global conservative features of the original method. Although, the time marching scheme of mesh variables is the same with the original method, the upwind fluxes are involved in the calculation of spatial derivatives, yielding a totally different approach from that of Chang's method. The upwind procedure breaks the space–time inversion invariance of the original scheme, so that the new scheme can be directly applied to capture discontinuities without spurious oscillations. In addition, the present method maintains low dissipation in a wide range of CFL number (from 10−6 to 1). Furthermore, we extend the upwind CE/SE method to solve the Euler equations by adopting three different approximate Riemann solvers including Harten, Lax and van Leer (HLL) Riemann solver, contact discontinuity restoring HLLC Riemann solver and mathematically rigorous Roe Riemann solver. Extensive numerical examples are carried out to demonstrate the robustness of the present method. The numerical results show that the new CE/SE solvers perform improved resolutions.

On some aspects of the discontinuous Galerkin finite element method for conservation laws

The paper is concerned with the numerical solution of nonlinear conservation laws and nonlinear convection–diffusion problems. We discuss two versions of this method: (a) Finite volume discontinuous Galerkin (FVDG) method, which is a generalization of the combined finite volume–finite element (FV–FE) method. Its advantage is the use of only one mesh (in contrast to the combined FV–FE schemes). However, it is of the first order only. (b) Further, the pure DGFE method of higher order is considered. In this case, a new limiting is developed to avoid spurious oscillations in the vicinity of shocks.

The Runge-Kutta Local Projection Discontinuous Galerkin Finite Element Method for Conservation Laws. IV: The Multidimensional Case

The Runge-Kutta Local Projection Discontinuous Galerkin Finite Element Method for Conservation Laws. IV: The Multidimensional Case

The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case

In this paper we study the two-dimensional version of the Runge-Kutta Local Projection Discontinuous Galerkin (RKDG) methods, already defined and analyzed in the one-dimensional case. These schemes are defined on general triangulations. They can easily handle the boundary conditions, verify maximum principles, and are formally uniformly high-ordrr accurate. Preliminary numerical results showing the performance of the schemes on a variety of initial-boundary value problems are shown.

TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: One-dimensional systems

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 u t + Σ i=1 d ( f i ( u) xi =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 1-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.

TVB Runge-Kutta Local Projection Discontinuous Galerkin Finite Element Method for Conservation Laws II: General Framework

TVB Runge-Kutta Local Projection Discontinuous Galerkin Finite Element Method for Conservation Laws II: General Framework

TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework

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 u t + ∑ i = 1 d ( f i ( u ) ) x i = 0 {u_t} + \sum \nolimits _{i = 1}^d {{{({f_i}(u))}_{{x_i}}} = 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.