Godunov's Scheme - Basic Scheme

Basic Scheme

Following the classical Finite-volume_method framework, we seek to track a finite set of discrete unknowns,

where the and form a discrete set of points for the hyperbolic problem:

If we integrate the hyperbolic problem over a control volume we obtain a Method_of_lines (MOL) formulation for the spatial cell averages:

which is a classical description of the first order, upwinded finite volume method. (c.f. Leveque - Finite Volume Methods for Hyperbolic Problems )

Exact time integration of the above formula from time to time yields the exact update formula:

Godunov's method replaces the time integral of for each

with a Forward Euler_method which yields a fully discrete update formula for each of the unkowns . That is, we approximate the integrals with

where is an approximation to the exact solution of the Riemman problem. For consistency, one assumes that

and that is increasing in the first argument, and decreasing in the second argument. For scalar problems where, one can use the simple Upwind_scheme, which defines .

The full Godunov scheme requires the definition of an approximate, or an exact Riemman solver, but in its most basic form, is given by: