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 solver

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[0,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.

Given the 2-wave nature of the HLL method, the method is ideal for an eigenvalue problem of dimension 2, that is hyperbolic systems that have 2 equations, such as the 1-dimensional shallow water equations. The HLL solver would not handle a higher dimensional system, such as the Euler equations containing 3 or more waves, accurately, given the entire region between sL*t and sR*t is averaged out. Thus, Toro and others developed the HLLC method, where "C" is for contact wave.

The HLLC solver

A similar algebraic analysis was derived for the conceptual paradigm described above, but now considering the presence of an additional wave within the region sL*t x sR*t.

Results and Discussion

The HLL solver performs adequately, closely resembling the Godunov + Exact Riemann Solver (ERS) over many problems. In situations where there is an isolated contact, or middle wave, the HLLC is by far the superior choice. These cases show the HLL solver is diffusive, especially when the contact moves slower than the grid speed.