Figure 2: Godunov’s scheme: local solutions of Riemann problems. At every interface, ,
and , a local Riemann problem is set up as a result of the discretization process
(bottom panel), when approximating the numerical solution by piecewise constant data. At time
these discontinuities decay into three elementary waves, which propagate the solution forward
to the next time level (top panel). The timestep of the numerical scheme must satisfy the
Courant–Friedrichs–Lewy condition, being small enough to prevent the waves from advancing more
than in .
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