This paper analyzes the convergence of discrete approximations to the linearized equations arising from an unsteady one-dimensional hyperbolic equation with a convex flux function. A simple modified Lax–Friedrichs discretization is used on a uniform grid, and a key point is that the numerical smoothing increases the number of points across the nonlinear discontinuity as the grid is refined. It ...

The monotonicity and stability of difference schemes for, in general, hyperbolic systems of conservation laws with source terms are studied. The basic approach is to investigate the stability and monotonicity of a non-linear scheme in terms of its corresponding scheme in variations. Such an approach leads to application of the stability theory for linear equation systems to establish stability ...

A conservative 3-D discontinuous Galerkin (DG) baroclinic model has been developed in the NCAR High-Order Method Modeling Environment (HOMME) to investigate global atmospheric flows. The computational domain is a cubed-sphere free from coordinate singularities. The DG discretization uses a high-order nodal basis set of orthogonal Lagrange-Legendre polynomials and fluxes of inter-element boundar...

More robust developments of schemes for hyperbolic systems, that avoid dependence upon a characteristic decomposition have been achieved by employing some form of a Lax-Friedrichs (LF)based flux. Such schemes permit the construction of higher order approximations without recourse to characteristic decomposition. This is achieved by using the maximum eigenvalue of the hyperbolic system within th...

The goal of this article is to design a new approximate Riemann solver for the two-layer shallow water system which is fast compared to Roe schemes and accurate compared to Lax-Friedrichs, FORCE, or GFORCE schemes (see [14]). This Riemann solver is based on a suitable decomposition of a Roe matrix (see [27]) by means of a parabolic viscosity matrix (see [16]) that captures some information conc...

We further developed a 2D spectral difference (SD) method code published in [1] and [2] to handle moving and deformable grids. The code is written using the spectral difference method where the solution points and flux points are arranged in a staggered fashion. We typically conduct computations using either 3rd-, 4th-, 5th-, or 6th-order SD method. The flow field within each cell is reconstruc...

In this paper we present the exact solution of the Riemann Problem for the non-linear shallow water equations with a step-like bottom. The solution has been obtained by solving an enlarged system obtained by adding an additional equation for the bottom geometry and then using the principles of conservation of mass and momentum across the step. The resulting solution is unique and satis es the p...

Many of the recently developed high-resolution schemes for hyperbolic conservation laws are based on upwind di erencing. The building block of these schemes is the averaging of an approximate Godunov solver; its time consuming part involves the eld-byeld decomposition which is required in order to identify the \direction of the wind." Instead, we propose to use as a building block the more robu...

A high resolution, second-order central difference method for incompressible flows is presented. The method is based on a recent second-order extension of the classic Lax-Friedrichs scheme introduced for hyperbolic conservation laws (Nessyahu H. & Tadmor E. (1990) J. Comp. Physics. 87, 408-463; Jiang G.-S. & Tadmor E. (1996) UCLA CAM Report 96-36, SIAM J. Sci. Comput., in press) and augmented b...

We present a new formulation of three-dimensional central finite volume methods on unstructured staggered grids for solving systems of hyperbolic equations. Based on the Lax-Friedrichs and Nessyahu-Tadmor one-dimensional central finite difference schemes, the numerical methods we propose involve a staggered grids in order to avoid solving Riemann problems at cell interfaces. The cells are baryc...