Results 1 
5 of
5
Nonoscillatory central differencing for hyperbolic conservation laws
 J. Comput. Phys
, 1990
"... Many of the recently developed highresolution 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 fieldbyfield decomposition which is required in orde ..."
Abstract

Cited by 233 (31 self)
 Add to MetaCart
Many of the recently developed highresolution 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 fieldbyfield 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 LaxFriedrichs (LxF) solver. The main advantage is simplicity: no Riemann problems are solved and hence fieldbyfield decompositions are avoided. The main disadvantage is the excessive numerical viscosity typical to the LxF solver. We compensate for it by using highresolution MUSCLtype 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.
Essentially nonoscillatory and weighted essentially nonoscillatory schemes for hyperbolic conservation laws
, 1998
"... In these lecture notes we describe the construction, analysis, and application of ENO (Essentially NonOscillatory) and WENO (Weighted Essentially NonOscillatory) schemes for hyperbolic conservation laws and related HamiltonJacobi equations. ENO and WENO schemes are high order accurate nite di ere ..."
Abstract

Cited by 178 (21 self)
 Add to MetaCart
(Show Context)
In these lecture notes we describe the construction, analysis, and application of ENO (Essentially NonOscillatory) and WENO (Weighted Essentially NonOscillatory) schemes for hyperbolic conservation laws and related HamiltonJacobi 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 selfcontained. 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 Of A Difference Scheme For Conservation Laws With A Discontinuous Flux
 SIAM J. Numer. Anal
, 1999
"... . Convergence is established for a scalar nite dierence scheme, based on the Godunov or EngquistOsher 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 ..."
Abstract

Cited by 64 (8 self)
 Add to MetaCart
(Show Context)
. Convergence is established for a scalar nite dierence scheme, based on the Godunov or EngquistOsher 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 Kruzkovtype 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...
Monotone Difference Approximations of BV Solutions to Degenerate ConvectionDiffusion Equations
 SIAM J. Numer. Anal
, 1998
"... We consider consistent, conservativeform, monotone difference schemes for nonlinear convectiondiffusion 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 ch ..."
Abstract

Cited by 30 (15 self)
 Add to MetaCart
(Show Context)
We consider consistent, conservativeform, monotone difference schemes for nonlinear convectiondiffusion 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 socalled 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...
Unconditionally Stable Explicit Schemes for the Approximation of Conservation Laws
"... We consider explicit schemes for homogeneous conservation laws which satisfy the geometric CourantFriedrichsLewy condition in order to guarantee stability but allow a time step with CFLnumber larger than one. A brief overview over existing unconditionally stable schemes for hyperbolic conservatio ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
(Show Context)
We consider explicit schemes for homogeneous conservation laws which satisfy the geometric CourantFriedrichsLewy condition in order to guarantee stability but allow a time step with CFLnumber 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 onedimensional scalar conservation laws with convex ux function and describe a possible way to extend the scheme to the twodimensional 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.