. Second-Order Approximation

Подпись: d2Ut dt2 Подпись: (1.24)

On replacing the spatial derivative of Eq. (1.15) by Eq. (1.22), the finite difference equation to be solved is

Eq. (1.24) is a second-order finite difference equation, the same order as the original partial differential equation. For a unique solution, two boundary conditions are required. This is given by the boundary conditions of the physical problem, Eq. (1.14); i. e.,

u0 = 0, uM = 0. (1.25)

On following Eq. (1.16), a separable solution of a similar form is sought,

U (t) = Re [Uee-lmt]. (1.26)

Подпись: Ui+1 + Подпись: ш2 (Ax)2 a2 Подпись: 2 Подпись: Hi + и i—1 — 0 Подпись: (1.27)

Substitution of Eq. (1.26) into Eqs. (1.24) and (1.25) leads to the following eigenvalue problem:

Подпись: (1.28)U0 = 0, UM = 0.

Подпись: r2 + Подпись: r + 1 = 0. Подпись: (1.29)

Two linearly independent solutions of finite difference equation (1.27) in the form of Eq. (1.4) can easily be found. The characteristic equation is

The two roots of Eq. (1.29) are complex conjugates of each other. The absolute value is equal to unity. Thus, the general solution of Eq. (1.27) may be written in the following form:

Подпись: (1.30)Ue = A sin (©l) + B cos (©l).

Подпись: © = cos Подпись: (1.31)


Upon imposition of boundary conditions (1.28), it is easy to find

B = 0, A sin (©M) = 0.

For a nontrivial solution, it is required that sin(0M) = 0. Hence,

&M = nn, n = 1, 2, 3,…

Подпись: nn M ■


This yields

Now, it is instructive to compare finite difference solutions (1.32) and (1.33) with the exact solution of the original partial differential equations (1.20) and (1.21). One obvious difference is that the exact solution has infinitely many eigenfrequencies and eigenfunctions, whereas the finite difference solution supports only a finite number (2M) of such modes. Furthermore, mn of Eq. (1.32) is a good approximation of the exact solution only for nn/M < 1. In other words, a second-order finite difference approximation provides good results only for the low-order long-wave modes. The error increases quickly as n increases.