In numerical analysis and computational fluid dynamics, Godunov's scheme is a conservative numerical scheme, suggested by Sergei 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.
Basic scheme Following the classical finite volume method framework, we seek to track a finite set of discrete unknowns,
Q i n = 1 Δ x ∫ x i − 1 / 2 x i + 1 / 2 q ( t n , x ) d x {\displaystyle Q_{i}^{n}={\frac {1}{\Delta x}}\int _{x_{i-1/2}}^{x_{i+1/2}}q(t^{n},x)\,dx}
where the x i − 1 / 2 = x low + ( i − 1 / 2 ) Δ x {\displaystyle x_{i-1/2}=x_{\text{low}}+\left(i-1/2\right)\Delta x} and t n = n Δ t {\displaystyle t^{n}=n\Delta t} form a discrete set of points for the hyperbolic problem:
q t + ( f ( q ) ) x = 0 , {\displaystyle q_{t}+(f(q))_{x}=0,}
where the indices t {\displaystyle t} and x {\displaystyle x} indicate the derivatives in time and space, respectively. If we integrate the hyperbolic problem over a control volume [ x i − 1 / 2 , x i + 1 / 2 ] , {\displaystyle [x_{i-1/2},x_{i+1/2}],} we obtain a method of lines (MOL) formulation for the spatial cell averages:
∂ ∂ t Q i ( t ) = − 1 Δ x ( f ( q ( t , x i + 1 / 2 ) ) − f ( q ( t , x i − 1 / 2 ) ) ) , {\displaystyle {\frac {\partial }{\partial t}}Q_{i}(t)=-{\frac {1}{\Delta x}}\left(f(q(t,x_{i+1/2}))-f(q(t,x_{i-1/2}))\right),}
which is a classical description of the first order, upwinded finite volume method. Exact time integration of the above formula from time t = t n {\displaystyle t=t^{n}} to time t = t n + 1 {\displaystyle t=t^{n+1}} yields the exact update formula:
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