Damping-Induced Numerical Instability
Now the phenomenon of “numerical instability” caused by excessive artificial damping is investigated. Suppose the solution of Eq. (7.21) is again computed with the initial condition (7.22), but with even larger damping, say RA1 = 15. Figure 7.6 shows the computed result as time increases. At t = 140, Figure 7.6a shows the onset of numerical instability. The instability wave has a wave length of about two mesh spacings or a Ax = n. This is grid-to-grid oscillation. At t = 150, Figure 7.6b shows that the numerical instability has grown in amplitude rapidly. At t = 400, Figure 7.6c shows that the instability completely overwhelms the numerical solution.
It is not difficult to understand why excessive artificial damping can lead to numerical instability. For this purpose, consider the Fourier-Laplace transform of Eq. (7.21). The transform of the finite difference equation is
_ _ 1 –
— i ши + i au =—- D (aAx)u.
Thus, the dispersion relation is
ш = a——— D (aAx).
Consider grid-to-grid oscillation instability wave with aAx = n. It is to be noted that a(n) = 0.0 (see Figure 2.3) and D (n) = 1.0 (see normalization condition (7.18)). Therefore, for this wave, dispersion relation (7.24) reduces to
-л – At
ш At = —i—. Ra
Figure 3.2 gives the stable region in the complex «At-plane. Along the imaginary axis, the four-time level DRP marching scheme is stable if Im(«At) > —0.29. Therefore, there will be numerical instability unless At/RA = 0.29. In the numerical example, the time step At is 0.02 and 1/RA = 15 so that At/RA = 0.3. Thus, the numerical scheme is outside the region of stability, and one should not be too surprised to see numerical instability.