Thus, its computing load is by a factor of 100 times larger than that in this article and its truncation error accumulation in the computational process is increased greatly; as well we have repeating computations of the classical FD scheme with first-order time accuracy on [ 0, t N ].
Though a POD-based reduced-order FD scheme with first-order time accuracy without adopting the extrapolation technique has been developed for NSIBEs (see [17]), it is also to test and verify only the comparison of solutions on the same time span [ 0, t N ] and it belongs to repeating computations.
The POD-based reduced-order FD extrapolating model with fully second-order accuracy here utilizes the given data (on the very short time span [ 0, t 0 ] and t 0 ≪ t N ) to predict future physic phenomena (on time span [ t 0, t N ] ) and has no repeating computations.
However, almost all existing POD-based reduced-order numerical methods employ the numerical solutions obtained from classical numerical methods on the total time span [ 0, t N ] to construct the POD basis and POD-based reduced-order models, and then recompute the solutions on the same time span [ 0, t N ], which is actually belong to repeating computations.
Especially, it has no repeating computations and uses the given solutions on the first fewer M time steps for Problem III to extrapolate other ((n-M)) solutions, which is completely different from existing reduced-order approaches (see, e.g., [14 35] etc).
However, almost all existing POD-based reduced-order numerical methods (see, e.g., [14 35]) employ numerical solutions obtained from classical numerical methods on the total time span ([0,T]) to form POD bases and establish reduced-order models, and then recompute the solutions on the same time span ([0,T]), which actually entails repeating computations on the same time span ([0,T]).
Remark 3.1 The system of equations (16) or (17) has no repeating computation and is different from the existing POD-based reduced-order numerical computational methods (see, e.g., [14 20, 22 31]) based on POD technique.
To avoid repeating computation of tensors [ Eq. (40)–(43)], we store them in a file.
A repeated computations in the optimization process, requires a large amount of computing time and resources.
It takes a total of 8,192 (8 × 8 × 8 × 16) searches, a tremendous computational load, to conduct a full search, i.e., repeated computations and comparisons in (2), for the identification of the optimal codevector.
Finally, the frequency-dependent terms are separated from the load vector to avoid repeated computations for different frequencies associated with the pseudo-excitations.
