In numerical analysis and computational fluid dynamics, Godunov's scheme is a conservative numerical scheme, suggested by Sergei Godunov in 1959,[1] 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.
Following the classical finite volume method framework, we seek to track a finite set of discrete unknowns,
{\displaystyle Q_{i}^{n}={\frac {1}{\Delta x}}\int _{x_{i-1/2}}^{x_{i+1/2}}q(t^{n},x)\,dx}
form a discrete set of points for the hyperbolic problem:
indicate the derivatives in time and space, respectively.
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.
[2] Exact time integration of the above formula from time
yields the exact update formula:
{\displaystyle Q_{i}^{n+1}=Q_{i}^{n}-{\frac {1}{\Delta x}}\int _{t^{n}}^{t^{n+1}}\left(f(q(t,x_{i+1/2}))-f(q(t,x_{i-1/2}))\right)\,dt.}
Godunov's method replaces the time integral of each
{\displaystyle \int _{t^{n}}^{t^{n+1}}f(q(t,x_{i-1/2}))\,dt}
with a forward Euler method which yields a fully discrete update formula for each of the unknowns
That is, we approximate the integrals with
is an approximation to the exact solution of the Riemann problem.
For consistency, one assumes 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 Riemann solver, but in its most basic form, is given by:
In the case of a linear problem, where
, and without loss of generality, we'll assume that
, the upwinded Godunov method yields:
which yields the classical first-order, upwinded Finite Volume scheme whose stability requires
Following Hirsch,[3] the scheme involves three distinct steps to obtain the solution at
, as follows: 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.