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