In numerical analysis and computational fluid dynamics, Godunov's scheme is a conservative numerical scheme, suggested by S. K. Godunov in 1959, for solving partial differential equations. One can think of this method as a conservative finite-volume method which solves exact, or approximate Riemann problems at each inter-cell boundary. In its basic form, Godunov's method is first order accurate in both space, and time, yet can be used as a base scheme for developing higher-order methods.
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Basic scheme
Following the classical Finite-volume method framework, we seek to track a finite set of discrete unknowns,
where the
If we integrate the hyperbolic problem over a control volume
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
Godunov's method replaces the time integral of each
with a Forward Euler method which yields a fully discrete update formula for each of the unknowns
where
and that
The full Godunov scheme requires the definition of an approximate, or an exact Riemann solver, but in its most basic form, is given by:
Linear problem
In the case of a linear problem, where
which yields the classical first-order, upwinded Finite Volume scheme whose stability requires
Three-step algorithm
Following Hirsch, the scheme involves three distinct steps to obtain the solution at
Step 1 Define piecewise constant approximation of the solution at
Step 2 Obtain the solution for the local Riemann problem at the cell interfaces. This is the only physical step of the whole procedure. The discontinuities at the interfaces are resolved in a superposition of waves satisfying locally the conservation equations. The original Godunov method is based upon the exact solution of the Riemann problems. However, approximate solutions can be applied as an alternative.
Step 3 Average the state variables after a time interval
The first and third steps are solely of a numerical nature and can be considered as a projection stage, independent of the second, physical step, the evolution stage. Therefore, they can be modified without influencing the physical input, for instance by replacing the piecewise constant approximation by a piecewise linear variation inside each cell, leading to the definition of second-order space-accurate schemes, such as the MUSCL scheme.