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## Godunov's scheme
## Additional recommended knowledgeFollowing
t = (n + 1)Δt. Since the piecewise constant approximation is an average of the solution over the cell size Δt, the spatial error is of order Δx, and hence the resulting scheme will be first-order accurate in space.
Note that this approximation corresponds to a finite volume method representation whereby the discrete values represent averages of the state variables over the cells. Exact relations for the averaged cell values can be obtained from the integral conservation laws.
t. The state variables obtained after Step 2 are averaged over each cell defining a new piecewise constant approximation resulting from the wave propagation during the time interval Δt. To be consistent, the time interval Δt should be limited such that the waves emanating from an interface do not interact with waves created at the adjacent interfaces. Otherwise the situation inside a cell would be influenced by interacting Riemann problems. This leads to the CFL condition | a_{max} | Δt < Δx / 2 where | a_{max} | is the maximum wave speed obtained from the cell eigenvalue(s) of the local Jacobian matrix.
The first and third steps are solely of a numerical nature and can be considered as a ## References- Godunov, S. K. (1959), "A Difference Scheme for Numerical Solution of Discontinuos Solution of Hydrodynamic Equations",
*Math. Sbornik*,**47**, 271-306, translated US Joint Publ. Res. Service, JPRS 7226, 1969. - Hirsch, C. (1990),
*Numerical Computation of Internal and External Flows*, vol 2, Wiley.
## Further reading- Laney, Culbert B. (1998),
*Computational Gas Dynamics*, Cambridge University Press. - Toro, E. F. (1999),
*Riemann Solvers and Numerical Methods for Fluid Dynamics*, Springer-Verlag. - Tannehill, John C., et al, (1997),
*Computational Fluid mechanics and Heat Transfer*, 2nd Ed., Taylor and Francis. - Wesseling, Pieter (2001),
*Principles of Computational Fluid Dynamics*, Springer-Verlag.
## See also |

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Godunov's_scheme". A list of authors is available in Wikipedia. |