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Central schemes

As before, we seek a piecewise-polynomial, tex2html_wrap_inline5425, which serves as an approximate solution to the exact evolution of sliding averages in (1.2.4),

Note that the polynomial pieces of tex2html_wrap_inline5393 are supported in the cells, tex2html_wrap_inline5445, with interfacing breakpoints at the half-integers gridpoints, tex2html_wrap_inline5447.

We recall that upwind schemes (1.2.5) were based on sampling (1.2.4) in the midcells, tex2html_wrap_inline5407. In contrast, central schemes are based on sampling (1.2.8) at the interfacing breakpoints, tex2html_wrap_inline5451, which yields
We want to utilize (1.2.9) in terms of the known cell averages at time level tex2html_wrap_inline5453. The remaining task is therefore to recover the pointvalues tex2html_wrap_inline5455, and in particular, the staggered averages, tex2html_wrap_inline5457. As before, this task is accomplished in two main steps:

Figure 1.2.2: Central differencing by Godunov-type scheme.

It is the staggered averaging over the fan of left-going and right-going waves centered at the half-integered interfaces, tex2html_wrap_inline5479, which characterizes the central differencing, consult Figure 1.2.2. A main feature of these central schemes - in contrast to upwind ones, is the computation of smooth numerical fluxes along the mid-cells, tex2html_wrap_inline5481, which avoids the costly (approximate) Riemann solvers. A couple of examples of central Godunov-type schemes is in order.

The first-order Lax-Friedrichs (LxF) approximation is the forerunner for such central schemes -- it is based on piecewise constant reconstruction, tex2html_wrap_inline5425 with tex2html_wrap_inline5485. The resulting central scheme, (1.2.12), then reads (with the usual fixed mesh ratio tex2html_wrap_inline5487)
Our main focus in the rest of this chapter is on non-oscillatory higher-order extensions of the LxF schemes.

next up previous contents
Next: Central schemes in one-space Up: A Short Guide to Previous: Upwind schemes

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