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3 Material Balance Finite Difference Equations in One Dimension

A widely used numerical method for approximating solutions to PDE problems is the finite difference method. To obtain a finite difference equation (FDE) for a partial differential equation, the continuous independent variables in the PDE (x and t in the previous examples) are restricted to a discrete grid of points, say in the x-direction and in the t-direction. Then the continuous derivatives in the PDE are approximated by difference quotients which involve the approximate solution to the PDE only at the grid points. The analytical solution techniques applied to PDE problems are generally based on calculus techniques and are not usually amenable to implementation on digital computers. In contrast, finite difference problems involve systems of algebraic equations and are well suited for use by a computational scientist.

There are two common methods for deriving finite difference equations. One method begins with a PDE and then uses finite difference formulas obtained from Taylor series expansions to approximate the various derivatives in the PDE. This approach is usually favored when performing a mathematical analysis of the consistency of the FDE with the PDE and convergence of the solution of the FDE to the PDE.

The second method for obtaining finite difference equations is based on a discrete conservation principle, similar to that introduced in Section 1. In this method, one need not begin with a PDE. All that is needed is a physical problem involving the density of some material and a rule for the flux of that quantity. In applications the quantity under consideration is usually mass, momentum or energy. The conservation law, or material balance, approach to obtaining finite difference equations proceeds by establishing a ``control interval'' about each of the spatial finite difference grid points. Finite difference equations then result from an application of the conservation principle of Section 1.