Finite difference schemes for differential equations

Abstract

(1) ~~~~~L = d p(x) d + q(x) and the related variational problem for the functional b ~~~~~~~~~~b (2) Q(u) = ] (pu'2 + qu') dx - 2 fu dx viz., min Q(u) uE Q where the class Q consists of smooth functions u(x) satisfying u(a) = u(b) = 0. Following Ritz, the solution of the variational problem may be discussed within the framework of the direct methods of the calculus of variations [1] by extending Q to the class of continuous functions with piecewise smooth derivatives. For purposes of deriving finite difference equations for the boundary value problem it is usual to consider continuous piecewise linear functions which reduce (2) to an easily evaluated sum, the Euler equations for which yield the difference equations. Thus, if p = 1, this results in approximating u by the second difference quotient (ui+l-2ui + ui_l)/Ax2. However, for problems with singular points this simple procedure may fail [10]. In this paper we illustrate certain theoretical and computational advantages which result for difference schemes by considering a canonical class of approximating functions chosen as piecewise smooth solutions of Lu = 0. In this case the resulting minimizing sequences for (2) lead, via the Euler equations for Q(u), to a system of difference equations Au' = where A = (Aij) is a symmetric, tri-diagonal matrix. We call difference equations derived in this manner patch equations. The solution, for a given subdivision of (a, b) by points xi , X2, , * Xn i iS X = (U(Xl), U(X2), u(x,)) where u(x) is the solution of Lu = f. Moreover, if K(x, y) is the Green's function for L on (a, b), so that LK = a(x - y), we also have in= K(xi, xj)Ajk = ik * Thus the structure of such difference equations parallels that of the differential equation.