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A Short Guide to Godunov-Type schemes

We want to solve the hyperbolic system of conservation laws
by Godunov-type schemes. To this end we proceed in two steps. First, we introduce a small spatial scale, tex2html_wrap_inline5377, and we consider the corresponding (Steklov) sliding average of tex2html_wrap_inline5379,
The sliding average of (1.2.1) then yields
Next, we introduce a small time-step, tex2html_wrap_inline5381, and integrate over the slab tex2html_wrap_inline5383,
We end up with an equivalent reformulation of the conservation law (1.2.1): it expresses the precise relation between the sliding averages, tex2html_wrap_inline5385, and their underlying pointvalues, tex2html_wrap_inline5379. We shall use this reformulation, (1.2.3), as the starting point for the construction of Godunov-type schemes.

We construct an approximate solution, tex2html_wrap_inline5389, at the discrete time-levels, tex2html_wrap_inline5391. Here, tex2html_wrap_inline5393 is a piecewise polynomial written in the form
where tex2html_wrap_inline5395 are algebraic polynomials supported at the discrete cells, tex2html_wrap_inline5397, centered around the midpoints, tex2html_wrap_inline5399. An exact evolution of tex2html_wrap_inline5389 based on (1.2.3), reads
To construct a Godunov-type scheme, we realize (1.2.4) -- or at least an accurate approximation of it, at discrete gridpoints. Here, we distinguish between the main methods, according to their way of sampling (1.2.4): these two main sampling methods correspond to upwind schemes and central schemes.

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