# Interdisciplinary Applied Mathematics

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(a)

FIGURE 17.13. (a) A weighting function that assigns different weighting coefficients for different snapshots. (b) Comparison of U-velocity in the main channel far upstream    of    the    intersection.    60    snapshots    are    used    and 3U+3V+3P    basis

functions are used in both methods.

accuracy.

In Figure 17.14(b), we compare the performance of weighted and standard KL techniques by fixing the number of snapshots and basis functions. The number of snapshots is fixed to 20, and the number of basis functions for U, V, and P is fixed to 3, i.e., (3U+3V+3P). The snapshots are eq-uispaced in time with a time period of 26.6 p,s between snapshots. The weighting coefficients for the weighted KL technique are computed using equation (17.22). The minimal weighting r is 1/6, and the steepness parameter is set to be 9. Figure 17.14(b) compares the weighted and standard KL techniques with the full transient simulation. The results indicate that the weighted KL basis is able to capture the velocity profile during the initial transient much more effectively than the standard KL approach. The steady-state solution predicted by both techniques is almost the same and compares well with the full transient simulation. From this we can conclude that with the same number of snapshots and basis functions, the weighted KL approach can offer better accuracy in resolving multiple time scales than the standard KL approach. In Figure 17.15, we compare the performance of weighted    and    standard KL    techniques    by    fixing    the    number    of    basis

functions (3U+3V+3P) and using different snapshots with each approach. The weighted KL method uses 22 snapshots to generate the basis functions (the weighting coefficients are again selected by the approach described in the previous paragraph), and the standard KL method uses 66 snapshots to generate basis functions. The result in Figure 17.15 indicates that the weighted KL technique offers better accuracy during the initial transient than the standard KL method, while both methods produce comparable accuracy at steady state. From this result, we can conclude that for a fixed number of basis functions, a weighted KL technique using fewer snapshots

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