(ii) Backward difference:

In this chapter we discuss the finite difference methods for linear partial differential equa-tions. We consider one example from each of Hyperbolic, Parabolic and Elliptic partial dif-ferential equations. The basic concept of consistency, stability and convergence of numerical schemes are discussed. 1 Hyperbolic equation Simplest hyperbolic partial differential equation is one way wave equation, (1.1) Pu = ut + aux = 0, − ∞ < x <∞, t> 0, u = u(x, t) with initial condition (1.2) u(x, 0) = u0(x). (1.1)- (1.2) is an initial value problem. First define the grid of points in the (x, t) plane by drawing vertical and horizontal lines through the points (xi, tn) x t grid linesgrid points where tn = n∆t, n = 0, 1, 2,... xi = ih, i = 0,±1,±2,... 1 The lines x = xi and t = tn are called grid lines and their intersections are called mesh points of the grid. We denote u(xi, tn) = u(ih, n∆t) = u n i The basic idea of finite difference method is to replace derivatives by finite differ-ences. This can be done in many ways; as example we hav