Decomposition-discontinuity method

One of the methods for the numerical solution of problems in mathematical physics. The term "decomposition-discontinuity" comes from gas dynamics. It denotes a process resulting from the contact of two masses of gas with different values of the dynamic parameters (density, velocity, pressure, internal energy). As applied to the numerical solution of problems in gas dynamics, the method is as follows. A difference grid is constructed for the domain where a problem is to be numerically solved (see Variable-grid method). It is assumed that the dynamic parameters of the gas within each cell are constant and equal to certain average values obtained from their known distribution. Then for every cell boundary of the grid a decomposition-discontinuity problem is solved for the two gas volumes in the adjacent cells. In the case of two semi-unbounded volumes of gas when the distribution of the dynamic parameters is constant in each of them and when the contact surface of the volumes is a plane, a solving algorithm can be compiled easily. The values of the dynamic parameters obtained from this solution and corresponding to the time-space position of the boundary separating the cells are taken for the values at the cell boundary. This approximation is valid at least within a time interval $\Delta t$, as long as the pairwise interactions do not affect each other. After computing the flows according to the values of the gas parameters at the cell boundaries and knowing the initial step distribution at time $t_0$, the step distribution at $t_0+\Delta t$ is calculated using the mass, momentum and energy balances for every cell of the difference grid. The difference grid itself can be changed during the calculations. Its motion can be either specified independently or defined in accordance with the characteristic features of the problem. The above limitation on $\Delta t$ is, essentially, the condition of stability of the computing scheme described.

This approach to constructing computation algorithms can be extended to problems of hydrodynamics with heat conduction, to elasticity theory, etc. Owing to an obvious physical interpretation and universality, and since the boundary conditions are adequate to the initial differential formulation, the decomposition-discontinuity method has become widely used for numerically solving problems in mathematical physics.

#### References[edit]

[1] |  L.D. Landau, E.M. Lifshitz, "Fluid mechanics" , Pergamon (1959) (Translated from Russian)

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[2] |  S. [S. Godunov] Godounov, et al., "Résolution numérique des problémes multidimensionells de la dynamique des gas" , MIR (1979) (Translated from Russian)