In this paper we discuss the issue of conservation and convergence to weak solutions of several global schemes, including the commonly used compact schemes and spectral collocation schemes, for solving hyperbolic conservation laws. It is shown that such schemes, if convergent boundedly ahnost everywhere, will converge to weak solutions. The results are extensions of the classical Lax-Wendroff t...

Abstract We consider numerical boundary conditions for high order finite difference schemes for solving convection-diffusion equations on arbitrary geometry. The two main difficulties for numerical boundary conditions in such situations are: (1) the wide stencil of the high order finite difference operator requires special treatment for a few ghost points near the boundary; (2) the physical bou...

We study the stability of some finite difference schemes for symmetric hyperbolic systems in two space dimensions. For the so-called upwind scheme and the Lax-Wendroff scheme with a stabilizer, we show that stability is equivalent to strong stability, meaning that both schemes are either unstable or `-decreasing. These results improve on a series of partial results on strong stability. We also ...

Two finite-difference methods for, geophysical .fluid problems are described, and stability conditions of these schemes arc discussed. These two schemes are formulated based upon a similar procedure given by Lax and Wendroff in order to obtain a second-order accuracy in finite-difference equations. However, the two schemes show remarkable differcnces.io their computational stability. One scheme...

The Lax-Wendroff theorem stipulates that a discretely conservative operator is necessary to accurately capture discontinuities. The discrete operator, however, need not be derived from the divergence form of the continuous equations. Indeed, conservation law equations that are split into linear combinations of the divergence and product rule form and then discretized using any diagonal-norm ske...

In this paper we present a new algorithm based on a Cartesian mesh for the numerical approximation of kinetic models on complex geometry boundary. Due to the high dimensional property, numerical algorithms based on unstructured meshes for a complex geometry are not appropriate. Here we propose to develop an inverse Lax-Wendroff procedure, which was recently introduced for conservation laws [21]...

In this paper, we explore the Lax-Wendroff (LW) type time discretization as an alternative procedure to the high order Runge-Kutta time discretization adopted for the high order essentially non-oscillatory (ENO) Lagrangian schemes developed in [2, 4]. The LW time discretization is based on a Taylor expansion in time, coupled with a local CauchyKowalewski procedure to utilize the partial differe...

Necessary and sufficient stability criteria for Friedrichs' scheme and the modified Lax-Wendroff scheme with smooth coefficients are derived by means of Kreiss' Matrix Theorem and the first Stability Theorem of Lax and Nirenberg. In this note we derive necessary and sufficient stability criteria for Friedrichs' scheme and the modified Lax-Wendroff scheme [8] for the hyperbolic system n (1) ". =...

We present a general technique to solve Partial Differential Equations, called robust stencils, which make them tolerant to soft faults, i.e. bit flips arising in memory or CPU calculations. We show how it can be applied to a two-dimensional Lax-Wendroff solver. The resulting 2D robust stencils are derived using an orthogonal application of their 1D counterparts. Combinations of 3 to 5 base ste...