Difference scheme conservative

B. van Leer, Towards the Ultimate Conservative Difference Scheme. V.A. Second-Order Sequel to Godunov s Method, J. Comput. Phys. 32 (1979). 

The above example shows that in designing difference schemes it is very desirable to reproduce the appropriate conservative law on a grid, The schemes with this property are said to be conservative. In subsequent sections the general method for constructing conservative schemes, which are convergent in the class of discontinuity coefficients, will be appreciated. Before we undertake the complete description of this method, it is worth noting two things. 

From a physical point of view, the finite difference method is mostly based based on the further replacement of a continuous medium by its discrete model. Adopting those ideas, it is natural to require that the principal characteristics of a physical process should be in full force. Such characteristics are certainly conservation laws. Difference schemes, which express various conservation laws on grids, are said to be conservative or divergent. For conservative schemes the relevant conservative laws in the entire grid domain (integral conservative laws) do follow as an algebraic corollary to difference equations. 

Substituting (12) and (13) into (11) and denoting by y,- the unknown function, we obtain the conservative difference scheme... 

Homogeneous difference schemes with weights. In a common setting it seems natural to expect that a difference scheme capable of describing this or that nonstationary process would be suitable for the relevant stationary process, that is, for du/dt = 0 we should have at our disposal a difference scheme from a family of homogeneous conservative schemes, whose use permits us to solve the equation Lu + / = 0. 

Equations of gas dynamics in integral form are aimed at designing conservative difference schemes by means of the integro-interpolation method ... 

In this context, it should be noted that conservative difference schemes may be good enough for the equation of the total energy, but approximate poorly the equation of the internal energy (14)... 

Samarskii, A. and Popov, Yu. (1969) Totally conservative difference schemes. Zh. Vychisl. Mat. i Mat. Fiz., 9, 953-954 (in Russian) English transl. in USSR Comput. Mathem. and Mathem. Physics. 

Chapters 2-5 are concerned with concrete difference schemes for equations of elliptic, parabolic, and hyperbolic types. Chapter 3 focuses on homogeneous difference schemes for ordinary differential equations, by means of which we try to solve the canonical problem of the theory of difference schemes in which a primary family of difference schemes is specified (in such a case the availability of the family is provided by pattern functionals) and schemes of a desired quality should be selected within the primary family. This problem is solved in Chapter 3 using a particular form of the scheme and its solution leads us to conservative homogeneous schemes. 

Gradient diffusion was assumed in the species-mass-conservation model of Shir and Shieh. Integration was carried out in the space between the ground and the mixing height with zero fluxes assumed at each boundary. A first-order decay of sulfur dioxide was the only chemical reaction, and it was suggested that this reaction is important only under low wind speed. Finite-difference numerical solutions for sulfur dioxide in the St. Louis, Missouri, area were obtained with a second-order central finite-difference scheme for horizontal terms and the Crank-Nicolson technique for the vertical-diffusion terms. The three-dimensional grid had 16,800 points on a 30 x 40 x 14 mesh. 

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