Time Step At: Accuracy Consideration

In Section 5.3, formulas to determine the largest time step, At, permitted by numer­ical stability requirement for waves governed by a given dispersion relation were developed. Here, the choice of At is revisited from the point of view of numerical accuracy. The At required by accuracy consideration may not be the same as that required by numerical stability. It is recommended that the smaller of the two At’s be used in actual computation.

From Figure 2.5, it is easy to find that, for the 7-point stencil DRP scheme, if the permissible numerical dispersion is limited to the range dal da < 0.003, the wave number a Ax of the numerical solution must not be larger than a* Ax (a* Ax = 0.9 from the figure). Also, from an enlarged Figure 3.1, it is easy to see that, if the range of mAt to be used is restricted, then it is possible to make both the numerical damping rate small (say, less than 1.2 x 10-5 or – Im(mAt) < 1.2 x 10-5) and that mAt and Re(mAt) differ by no more than 0.0002. This is accomplished if mAt is limited to less than m* At (m*At = 0.19 from the figure). To keep the computation to within this accuracy in the wave number as well as the angular frequency space, the size of At must be chosen appropriately. From dispersion relation (5.40), it is easily found that

Now, to maintain numerical accuracy to da |da < 0.003, it is necessary to keep a Ax and в Ay smaller than a*Ax = 0.9. By Eq. (5.56), this yields

The largest value of mAt that complies with the this accuracy is mAt = 0.19. Therefore, by Eq. (5.57), At must be such that

Eq. (5.58) is the formula for At if this accuracy is to be guaranteed in the computation. The requirement on At for numerical stability, from Eq. (5.45), is

The right side of Eq. (5.59) is slightly larger than that of Eq. (5.58). In other words, the requirement for numerical accuracy is slightly more stringent than that for numerical stability. Eq. (5.58) should be used for determining the step size At in a numerical simulation. Note that the choice of Eq. (5.58) ensures that the dispersion rela­tion matches exactly the wave number and angular frequency range of acceptable accuracy.