Many of the recently developed high-resolution schemes for hyperbolic conservation laws are based on upwind differencing. The building block of these schemes is the averaging of an approximate Godunov solver; its time consuming part involves the field-by-field decomposition which is required in order to identify the “direction of the wind. ” Instead, we propose to use as a building block the more robust Lax-Friedrichs (LxF) solver. The main advantage is simplicity: no Riemann problems are solved and hence field-by-field decompositions are avoided. The main disadvantage is the excessive numerical viscosity typical to the LxF solver. We compensate for it by using high-resolution MUSCL-type interpolants. Numerical experiments show that the quality of the results obtained by such convenient central differencing is comparable with those of the upwind schemes. c○Academic Press, Inc.

In these lecture notes we describe the construction, analysis, and application of ENO (Essentially Non-Oscillatory) and WENO (Weighted Essentially Non-Oscillatory) schemes for hyperbolic conservation laws and related Hamilton-Jacobi equations. ENO and WENO schemes are high order accurate nite di erence schemes designed for problems with piecewise smooth solutions containing discontinuities. The key idea lies at the approximation level, where a nonlinear adaptive procedure is used to automatically choose the locally smoothest stencil, hence avoiding crossing discontinuities in the interpolation procedure as much as possible. ENO and WENO schemes have been quite successful in applications, especially for problems containing both shocks and complicated smooth solution structures, such as compressible turbulence simulations and aeroacoustics. These lecture notes are basically self-contained. It is our hope that with these notes and with the help of the quoted references, the readers can understand the algorithms and code

. Convergence is established for a scalar nite dierence scheme, based on the Godunov or Engquist-Osher ux, for scalar conservation laws having a ux that is spatially dependent through a possibly discontinuous coecient. Other works in this direction have established convergence for methods employing the solution of 2x2 Riemann problems. The algorithm discussed here uses only scalar Riemann solvers. Satisfaction of a set of Kruzkov-type entropy inequalities is established for the limit solution, from which geometric entropy conditions follow. Assuming a piecewise constant coecient, it is shown that these conditions imply L 1 -contractiveness for piecewise C 1 solutions, thus extending a well known theorem. Key words. conservation laws, dierence approximations, discontinuous coecients AMS subject classications. 35L65, 65M06, 65M12, 35R05 1. Introduction. The subject of this paper is a nite dierence algorithm for computing approximate solutions of the Cauchy problem for scala...

We consider consistent, conservative-form, monotone difference schemes for nonlinear convection-diffusion equations in one space dimension. Since we allow the diffusion term to be strongly degenerate, solutions can be discontinuous and are in general not uniquely determined by their data. Here we choose to work with weak solutions that belong to the BV (in space and time) class and, in addition, satisfy an entropy condition. A recent result of Wu and Yin [31] states that these so-called BV entropy weak solutions are unique. The class of equations under consideration is very large and contains, to mention only a few, the heat equation, the porous medium equation, the two phase ow equation and hyperbolic conservation laws. The difference schemes are shown to converge to the unique BV entropy weak solution of the problem. In view of the classical theory for monotone difference approximations of conservation laws, the main difficulty in obtaining a similar convergence theory in the present cont...

We consider explicit schemes for homogeneous conservation laws which satisfy the geometric Courant-Friedrichs-Lewy condition in order to guarantee stability but allow a time step with CFL-number larger than one. A brief overview over existing unconditionally stable schemes for hyperbolic conservation laws is provided, although the focus is on LeVeque's large time step Godunov scheme. For this scheme we explore the question of entropy consistency for the approximation of one-dimensional scalar conservation laws with convex ux function and describe a possible way to extend the scheme to the two-dimensional case. Numerical calculations and analytical results show that an increase of accuracy can be obtained because the error introduced by the modi ed evolution step of the large time step Godunov scheme may be less important than the error due to the projection step.