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Upwind schemes

Let tex2html_wrap_inline5403 abbreviates the cell averages, tex2html_wrap_inline5405. By sampling (1.2.4) at the mid-cells, tex2html_wrap_inline5407, we obtain an evolution scheme for these averages, which reads
Here, it remains to recover the pointvalues, tex2html_wrap_inline5409, in terms of their known cell averages, tex2html_wrap_inline5411, and to this end we proceed in two steps:

The solution of (1.2.7) is composed of a family of nonlinear waves - left-going and right-going waves. An exact Riemann solver, or at least an approximate one is used to distribute these nonlinear waves between the two neighboring cells, tex2html_wrap_inline5419 and tex2html_wrap_inline5421. It is this distribution of waves according to their direction which is responsible for upwind differencing, consult Figure 1.2.1. We briefly recall few canonical examples for this category of upwind Godunov-type schemes.

Figure 1.2.1: Upwind differencing by Godunov-type scheme.

The original Godunov scheme is based on piecewise-constant reconstruction, tex2html_wrap_inline5423, followed by an exact Riemann solver. This results in a first-order accurate upwind method [11], which is the forerunner for all other Godunov-type schemes. A second-order extension was introduced by van Leer [19]: his MUSCL scheme reconstructs a piecewise linear approximation, tex2html_wrap_inline5425, with linear pieces of the form tex2html_wrap_inline5427 so that tex2html_wrap_inline5429. Here the tex2html_wrap_inline5431-s are possibly limited slopes which are reconstructed from the known cell-averages, tex2html_wrap_inline5433. (Throughout this lecture we use primes, tex2html_wrap_inline5435, to denote discrete derivatives, which approximate the corresponding differential ones). A whole library of limiters is available in this context, so that the co-monotonicity of tex2html_wrap_inline5393 with tex2html_wrap_inline5439 is guaranteed, e.g., [42]. The Piecewise-Parabolic Method (PPM) of Colella-Woodward [6] and respectively, ENO schemes of Harten [13], offer, respectively, third- and higher-order Godunov-type upwind schemes. (A detailed account of ENO schemes can be found in lectures of C.W. Shu in this volume). Finally, we should not give the impression that limiters are used exclusively in conjunction with Godunov-type schemes. The positive schemes of Liu and Lax, [27], offer simple and fast upwind schemes for multidimensional systems, based on an alternative positivity principle.

next up previous contents
Next: Central schemes Up: A Short Guide to Previous: A Short Guide to

Eitan Tadmor
Mon Dec 8 17:34:34 PST 1997