The Short Wave Component of Finite Difference Schemes

It was pointed out previously that when a central finite difference scheme was used to approximate the spatial derivative, the wave number of the finite difference scheme a was related to the actual wave number a. Figure 2.1 shows a typical relationship between a Ax and a Ax. Earlier, it was also suggested that the wave spectrum might be divided into two parts; the long waves (waves for which a ~ a ) and the short waves (waves for which a is very different from a). The long waves behave like the corresponding wave component of the exact solution. In this chapter, attention is focused on the short waves.

7.1 The Short Waves

To fix ideas, consider the initial value problem associated with the linearized Euler equations in one space dimension without mean flow. The same dimensionless vari­able as in Section 6.4 is used. The dimensionless linearized momentum and energy equations are as follows:

d u d p + = 0 91 dx	(7.1)
9 p d u — +—— = 0. dt dx	(7.2)
For simplicity, consider the following initial conditions,
t = 0, u = 0	(7.3)
P = f (x)-	(7.4)
It is easy to verify that the exact solution of (7.1) to (7.4) is
1 P = 2 [f (x -1) + f (x +t)] –	(7.5)
This solution suggests that half the initial pressure pulse propagates to the left and the other half propagates to the right at the speed of sound (unity in dimensionless units). Now consider a “boxcar” initial condition as follows:
f (x) = H(x + M) – H(x – M),	(7.6)

where M is a large positive number and H(x) is the unit step function. The wave num­ber spectrum of Eq. (7.6) extends well beyond the range – п < a < n. A considerable fraction of the spectrum falls in the short wave range. This offers an excellent exam­ple on the wave propagation characteristics of the short waves of finite difference schemes. From Eq. (7.5), the exact solution is

1 1

p(x, t) = 2 [H(x – t + M) – H(x – t – M)] + 2 [H(x +1 + M) – H(x +1 – M)].

(7.7)

It is of interest to find out what happens when the problem is solved by the 7-point stencil dispersion-relation-preserving (DRP) scheme. On discretizing Eqs. (7.1) and (7.2) according to the DRP scheme, the resulting finite difference equations are

3

E"’ = – £ ajP“j

j=-3

3

F“ = – £ aUH,

j=-3

3

u("+1) = uf + AtJ2biEf – j)

j=0

3

рГ1’=p»+1>+a, j2i>,f;"-h.

j=0

The initial conditions corresponding to Eqs. (7.3) and (7.6) are

U" = 0, " < 0

(") = 1H(l + M) – H(l – M), " = 0

Pl {0, " < 0

Figure 7.1 shows the numerical solution with the boxcar initial condition

(M = 50). As can be seen, the solution is badly contaminated by short waves. The lead waves are the grid-to-grid oscillatory waves. The envelope of the amplitude of the short waves oscillates spatially (giving the appearance of lumps of waves). The longer dispersive waves are trailing the main pulse solution.

This example clearly shows that short waves are pollutants of numerical solu­tions. They can be generated by discontinuous initial or boundary conditions. To render numerical solutions acceptable, a way must be found to suppress or eliminate the short waves without interfering with the long waves.