Multidimensional Waves

The analysis of the previous sections is primary for one-dimensional waves. However, it can easily be generalized to waves in two or three dimensions.

Suppose the multidimensional dispersion relation of a wave system is

Подпись: u(xi, t) Подпись: (4.37)

Again, assuming there is a simple pole on the real «-axis given by«(ai), it is straight­forward to obtain by the Residue Theorem that the contribution of this pole to the solution is

Подпись: u(xi, t) Подпись: (4.38)

The triple integral may be evaluated asymptotically for large t by the method of stationary phase in multidimensions. Thus,

Подпись: xi 9« t dai ’ Подпись: i = 1, 2, 3. Подпись: (4.39)


Подпись: V = ^ i 9a ’ Подпись: i = 1, 2, 3. Подпись: (4.40)

In Eq. (4.38), det is the determinant of and g depends on the number of factors of n/4 arising from path rotation. Now Eq. (4.39) gives the group velocity, Vi, in three dimensions as follows:

It is easy to see, for a multidimensional finite difference system, that the same analysis applies. Therefore, the conclusions concerning numerical dispersion and dissipation
obtained for a one-dimensional system are also true for multidimensional finite difference approximations.

EXERCISES 4.1. Suppose the initial value problem,

^ + c^r = 0, u(x, 0) = f (x),

dt dx

is computed by approximating the du/dx term by a finite difference quotient, show

(i) If the standard sixth-order central difference approximation is used, the numerical solution can only have trailing waves.

(ii) If the 7-point stencil DRP scheme is used, the numerical solution may have both leading and trailing waves. Estimate the wavelength of the leading waves.

4.2. In solving the convective wave equation,

d U d U

—– + C = 0,

d t dx

by approximating the spatial derivative by finite difference quotient and marching the solution in time by the fourth-order RK scheme, it has been shown that the solution, in wave number space, is governed by the equation


Подпись: U(n+1) = U(n)1 + ^2 Cj (-icaAt)), j=1

where a(a) is the wave number of the finite difference scheme and n is the time level. One may consider this as a first-order finite difference equation in n (see Chap­ter 1). Find the exact solution of this finite difference equation. To avoid numerical instability, the absolute value of the characteristic root must be smaller than 1.0. Suppose a is the wave number of the standard sixth-order central difference scheme and Cj’s are the coefficients of the standard fourth-order RK scheme. Determine the largest time step At that may be used without encountering numerical instability.