48 CHAPTER 5. NUMERICAL MODEL
condition is that it ensures that the error on the approximation of the time-derivative is
smaller than the error on the approximation of the spatial-derivatives. Consequently, the
value of CFLmax depends on the specic solver and on the order of the basis functions.
For fourth-order basis functions and the generalized alpha solver 47, Ref. 48 reports a
value of CFL4th
max = 0:05, which is an empirical result for a specic model.
To determine a reasonable trade-o between numerical accuracy and computational
time, we study the convergence of the transient solution towards the steady solution for
dierent values of the temporal resolution t0=t. The acoustic energy Eac(t) is shown in
Fig. 5.3(a)
for dierent values of t and normalized by the steady time-averaged energy
Efd
of the frequency-domain calculation, and it is thus expected to converge to the
ac(1)
Efd
unity for long times. In Fig. 5.3(b), Eac(1000t0)=
is plotted versus the temporal
ac(1)
resolution t0=t, which shows how the accuracy of the time-domain solution increases as
the temporal resolution is increased. In Ref. 30 Appendix E the time step of t = t0=256,
the circled point in Fig. 5.3(b), was chosen as a reasonable trade-o, for which the timedomain
energy converge to 99.4% of the energy of the steady calculation.
In Ref. 30 Appendix E it was further noted that the fastest convergence was obtained
when actuating the system at its (numerically determined) resonance frequency fres. When
shifting the actuation frequency half the resonance width 1
2f away from fres, the energy
Eac(t) for t = t0=256 converged to only 95% of the steady value
Efd
(calculated in
ac(1)
the frequency domain), thus necessitating smaller time steps to obtain reasonable convergence.