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# Approximations for the Godunov Scheme

An approximation scheme for the Godunov method would either be one that a) approximates the values of p and u in the star region, from which the numerical flux can be directly computed, or b) approximates the numerical flux itself. The approximate Riemann solvers we learned about previously, on my wiki page on PVRS, TRRS, TSRS, and the like, are approximation methods of type 'a'. The HLL and HLLC solvers, as we will see here, are approximation techniques of type 'b'.

# The HLL and HLLC solvers

The HLL solver was developed first, and consists of the following conceptual paradigm. At the interface of any 2 cells, one imagines the generation of 2 outgoing waves, traveling to the left and to the right. These outgoing waves travel with speeds SL and SR. Their speed*time define a certain distance, within which a perturbative signal can travel into the surrounding fluid medium. Thus, if you consider a large enough volume that contains these waves, given by [xL,xR]x[o,T], where xL is the left spatial boundary, xR is the right spatial boundary, and 0-T is the time interval, you can do an algebraic analysis of the conservation laws to solve for the flux of the fluid variables across these waves.

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