Godunov's scheme

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.

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

where the ${\displaystyle x_{i}=x_{\text{low}}+\left(i-1/2\right)\Delta x}$ and ${\displaystyle t^{n}=n\Delta t}$ form a discrete set of points for the hyperbolic problem:

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