Version 6 (modified by 12 years ago) ( diff ) | ,
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# Big Picture

- Define a piece-wise function of initial data; do this by computing cell averages: U(i,n) = (1/delta_x)*Integral[u(x,tn)]dx evaluated over the entire cell (i.e. over flanking intercell boundaries)

- Solve the IVP for the original conservation law, but with modified (discretized) initial data to get the solution for the next time level

- This produces local Riemann Problems (RPs), centered on intercell boundaries.

- Solve these local RP's, and use solution to compute fluxes

## The method

In conservative form, the Godunov method is written:

where F(i-½) is the numerical flux, which = F(u(i+½, 0)) (i.e. the physical flux evaluated using the solution to local RP)

where delta_t is constrained by the delta_t ⇐ delta_x/max[S(n)] ← here s(n) is a given wave speed. note this allows a wave to travel a full delta_x in a time-step

we don't worry about wave-wave interaction based on linearity — that is, assume wave interactions do not lead to wave accelerations

we can specify the time constraint as follows, delta_t = cfl*delta_x/max[s(n)], if we let 0<cfl⇐1. a cfl = 1 gives the most efficient time marching scheme (the largest time steps). max[s(n)] finds the max speed on the entire domain.

u(i, n+1) = average over the cell

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