# Godunov's scheme

**Godunov's scheme**is aconservative numerical scheme , suggested by S. K. Godunov in 1959, for solvingpartial differential equations . In this method, the conservative variables are considered aspiecewise constant over the mesh cells at each time step and the time evolution is determined by the exact solution of theRiemann problem (shock tube) at the inter-cell boundaries (Hirsch, 1990).Following "Hirsch", the scheme involves three distinct steps to obtain the solution at $t\; =\; (n+1)\; Delta\; t\; ,$ from the known solution at $\{t\; =\; n\; Delta\; t\}\; ,$, as follows:

**"Step 1**" Define piecewise constant approximation of the solution at $\{t\; =\; (n+1)\; Delta\; t\}\; ,$. Since the piecewise constant approximation is an average of the solution over the cell of size $\{Delta\; x\}\; ,$, the spatial error is of order $\{Delta\; x\}\; ,$, and hence the resulting scheme will be first-order accurate in space.Note that this approximation corresponds to afinite 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 $\{Delta\; t\}\; ,$. 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 $\{Delta\; t\}\; ,$. To be consistent, the time interval $\{Delta\; t\}\; ,$ 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 $|\; a\_\{max\}\; |\; Delta\; t\; <\; Delta\; x/2\; ,$ 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 "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.**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.**ee also***

Godunov's theorem

*High resolution scheme

*MUSCL scheme

*Sergei K. Godunov

*Total variation diminishing

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**Godunov**— is a Russian name. Godunov can refer to the following: * Boris Godunov a regent of Russia from 1584 to 1598 * Sergei K. Godunov a Russian born mathematician who contributed to Finite volume method ** Godunov s scheme, a mathematical method… … Wikipedia**Godunov's theorem**— Godunov s theorem, also known as Godunov s order barrier theorem, is a mathematical theorem important in the development of the theory of high resolution schemes for the numerical solution of partial differential equations.Professor Sergei K.… … Wikipedia**MUSCL scheme**— MUSCL stands for Monotone Upstream centered Schemes for Conservation Laws , and the term was introduced in a seminal paper by Bram van Leer (van Leer, 1979). In this paper he constructed the first high order , total variation diminishing (TVD)… … Wikipedia**Sergei K. Godunov**— Infobox Person name = Sergei Konstantinovich Godunov caption = Sergei Konstantinovich Godunov birth date = birth date and age|1929|07|17 birth place = Moscow death date = death place = Sergei Konstantinovich Godunov (b. July 17, 1929 in Moscow,… … Wikipedia**High-resolution scheme**— High resolution schemes are used in the numerical solution of partial differential equations where high accuracy is required in the presence of shocks or discontinuities. They have the following properties:*Second or higher order spatial accuracy … Wikipedia**Bram van Leer**— Infobox Person name = Bram van Leer caption = Bram van Leer birth date = birth place = The Netherlands death date = death place = Bram van Leer is professor of aerospace engineering at the University of Michigan, in Ann Arbor. He specialises in… … Wikipedia**List of numerical analysis topics**— This is a list of numerical analysis topics, by Wikipedia page. Contents 1 General 2 Error 3 Elementary and special functions 4 Numerical linear algebra … Wikipedia**List of Russian mathematicians**— Andrey Kolmogorov, a preeminent 20th century mathematician. This list of Russian mathematicians includes the famous mathematicians from the Russian Empire, the Soviet Union and the Russian Federation. This list is incomplete; you can help by … Wikipedia**List of mathematics articles (G)**— NOTOC G G₂ G delta space G networks Gδ set G structure G test G127 G2 manifold G2 structure Gabor atom Gabor filter Gabor transform Gabor Wigner transform Gabow s algorithm Gabriel graph Gabriel s Horn Gain graph Gain group Galerkin method… … Wikipedia**Crank–Nicolson method**— In numerical analysis, the Crank–Nicolson method is a finite difference method used for numerically solving the heat equation and similar partial differential equations.[1] It is a second order method in time, implicit in time, and is numerically … Wikipedia