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Godunov's scheme

Godunov's scheme is a conservative numerical scheme, suggested by S. K. Godunov in 1959, for solving partial differential equations. In this method, the conservative variables are considered as piecewise constant over the mesh cells at each time step and the time evolution is determined by the exact solution of the Riemann problem (shock tube) at the inter-cell boundaries (Hirsch, 1990).

Following "Hirsch", the scheme involves three distinct steps to obtain the solution at $t = \left(n+1\right) Delta t ,$ from the known solution at $\left\{t = n Delta t\right\} ,$, as follows:

"Step 1" Define piecewise constant approximation of the solution at $\left\{t = \left(n+1\right) Delta t\right\} ,$. Since the piecewise constant approximation is an average of the solution over the cell of size $\left\{Delta x\right\} ,$, the spatial error is of order $\left\{Delta x\right\} ,$, 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.

"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 $\left\{Delta t\right\} ,$. 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 $\left\{Delta t\right\} ,$. To be consistent, the time interval $\left\{Delta t\right\} ,$ 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 $| a_\left\{max\right\} | Delta t < Delta x/2 ,$ where $| a_\left\{max\right\} | ,$ 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 "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.

References

* Godunov, S. K. (1959), "A Difference Scheme for Numerical Solution of Discontinuous 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.

* 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.

ee also

* Godunov's theorem
* High resolution scheme
* MUSCL scheme
* Sergei K. Godunov
* Total variation diminishing

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