ABSTRACT
The basic concept in finite difference methods, as applied to boundary-value problems, is the representation of governing differential equations and associated boundary conditions by appropriate finite difference equations. This replacement is accomplished by approximating derivatives in the differential equations with finite difference quotients that are combinations of dependent (unknown) function values at specified values of the independent variables. By writing the difference equations at specified values of the independent variables, we are led to systems of simultaneous algebraic equations that may be solved by elementary means with the aid of high-speed computers. Accordingly, we interpret a finite difference method as a numerical procedure that approximates known exact 1 differential equations and boundary conditions—say, of an elasticity problem. Then we solve the resulting approximate equations exactly or approximately. On the other hand, we shall see in Chapter 4 that finite element methods approximate the elastic continua by assemblages of discrete elastic systems. Then we solve the resulting discrete systems exactly or approximately.
