Abstract
This chapter summarizes the application of the original Godunov method to the one-dimensional, scalar conservation laws and illustrates the applications with computational examples. In the original Godunov method, the right and left states of the Riemann problem across the interface are taken equal to the average values over the cells. The application to the Lighthill, Whitham, and Richards (LWR) model is also investigated with the description of the method for the Buckley-Leverett equation. The non-convex nature of the flux function in the Buckley-Leverett equation induces particular problems that have consequences on the stability criterion. The essential point in Godunov-type schemes is the computation of the fluxes at the interfaces between the computational cells. When the conservation law is non-linear, the computation of the fluxes at the interfaces of the left and right hand boundaries of the domain proves to be more complicated. Since the direction of the characteristics depends on the value of the flow variable, it is not always possible to prescribe the desired value of the flux or the variable.
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