Asymptotic Behavior of Solutions to the Finite-Difference Wave Equation

where up in Eq. (2) corresponds to u(jôx, not) in Eq. (1), and where a = cSt/Sx. Here öt and Sx are the time and space intervals, respectively. We consider the case — co < x < », í > 0. It was shown by Courant, Friedrichs, and Lewy in a wellknown paper [1] that if up and u,x are prescribed for ally, then the computational process represented by Eq. (2) will yield values for w/ which converge to the exact solution of Eq. (1) as 5x —> 0 (for corresponding initial conditions), provided that the stability condition a < 1 is maintained. They point out that if a > 1, then the procedure (2) cannot possibly be convergent, because the domain of dependence of a point (a;, t) via Eq. (2) does not include all points of its domain of dependence via Eq. (1), and consequently a change in initial values near the endpoints of the domain of dependence will affect the solution of Eq. (1) but not that of Eq. (2). Since the proof of reference (1) is nonconstructive, it does not shed light on two aspects of the stable case a < 1. The first of these is that the speed with which signals propagate under Eq. (2) is c/a, which in fact —» a> as a —> 0; on the other hand, the signal speed associated with Eq. (1) is c. A second and related apparent paradox arises from the converse argument to that used by the above authors; since for a < 1 the domain of dependence for Eq. (2) includes points not in the domain of dependence for Eq. (1), it should be possible to alter the initial conditions so as to affect the solution of Eq. (2) but not that of Eq. (1). Of course, the fact that convergence does ensue as ôx —» 0, for any choice of a < 1, demonstrates indirectly that these two apparent discrepancies must become unimportant in some sense as Sa; —► 0; nevertheless, it is of interest to demonstrate this result directly, and also to obtain explicit formulas for the errors. We first obtain an exact solution of Eq. (2), corresponding to the initial conditions Mo° = A, íío1 = A + Bôt with uP = Uj1 = 0 for all j ^ 0. Any more general initial conditions can be constructed by means of a superposition of conditions of this form. We then consider the case a <K 1, which accentuates the apparent paradox, and use the saddle point method to obtain explicit asymptotic expressions for u¡n