Finite difference techniques mathematical formulation

Without a solution, formulated mathematical systems (models) are of little value. Four solution procedures are mainly followed the analytical, the numerical (e.g., finite different, finite element), the statistical, and the iterative. Numerical techniques have been standard practice in soil quality modeling. Analytical techniques are usually employed for simplified and idealized situations. Statistical techniques have academic respect, and iterative solutions are developed for specialized cases. Both the simulation and the analytic models can employ numerical solution procedures for their equations. Although the above terminology is not standard in the literature, it has been used here as a means of outlining some of the concepts of modeling. 

Mathematical formulations of various boundary conditions were discussed in Section 2.3. These boundary conditions may be implemented numerically within the finite volume framework by expressing the flux at the boundary as a combination of interior values and boundary data. Usually, boundary conditions enter the discretized equations by suppression of the link to the boundary side and modification of the source terms. The appropriate coefficient of the discretized equation is set to zero and the boundary side flux (exact or approximated) is introduced through the linearized source terms, Sq and Sp. Since there are no nodes outside the solution domain, the approximations of boundary side flux are based on one-sided differences or extrapolations. Implementation of commonly encountered boundary conditions is discussed below. The technique of modifying the source terms of discretized equation can also be used to set the specific value of a variable at the given node. To set a value at... 

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