In this section, the error estimates of the solutions of the POD-based reduced-order FD extrapolating model are first provided for guidance to choose the number of POD basis functions and renew the POD basis, and then the algorithm implementation for the POD-based reduced-order FD extrapolating model is given.
4.1 Error estimates and criterion of renewing POD basis
It is obvious that when , by (13), (12) can be written as the following form like (1)-(4):
(14)
(15)
(16)
(17)
where
It is obvious that the stability condition of (14)-(17) is also and .
Equations (12) and (13) can also be rewritten as vector form denoted by
(18)
Let . Subtracting (11) and (13) from (9) yields
(19)
Subtracting (18) from (9) yields
(20)
By (8) and (19), we have the following error estimates:
(21)
Let M = max{, , }. Based on the stability conditions of (1)-(4) and (14)-(17), we have . Therefore, from (1)-(4) and (14)-(17), we obtain
(22)
Summing (22) from L to yields
(23)
Put . Then, from (23), we obtain and (). Thus, we have from (21) that
(24)
Since the absolute value of each component for vector is not more than its norm, by combining Theorem 1 with (21) and (24), we have the following result.
Theorem 2 Based on the stability conditions , , and of (1)-(4) and (14)-(17), the error estimates between the solution for NSIBEs and the solutions obtained from the POD-based reduced-order FD extrapolating model (11)-(13) are denoted by
(25)
where (), (), and M = max{, , }.
Remark 2 The error estimates in Theorem 2 provide guidance for choosing the number of POD basis functions, namely, we should take , , , and such that . () are caused by extrapolating iteration and may act as the criterion for renewing the POD basis, namely, if , the old POD basis is substituted with the new POD basis regenerated from new snapshots.
4.2 Algorithm implementation for the POD-based reduced-order FD extrapolating model
The algorithm implementation for the POD-based reduced-order FD extrapolating model (11)-(13) consists of the following five steps.
Step 1. For given the Reynolds number Re, the Prandtl number Pr, boundary value functions , , and , initial value functions , , and , the time step increment Δt, and the spatial step increments Δx and Δy, let , , and , solving the following classical FD scheme, for :
yields L (with usually ) groups of classical FD solutions , , , and (, , ), further, constructing a set of snapshots () with elements (for actual engineering problems, the ensemble of snapshots is obtained from physical system trajectories by drawing samples from experiments), where , , , and (, , , , ), respectively.
Step 2. Form the snapshot matrices () and solve the linear systems of equations obtaining the eigenvalues (, ) and corresponding eigenvectors (, ).
Step 3. For the error needed, determine the numbers (, ) of POD basis functions such that , and construct the POD basis (where , , ).
Step 4. Write , , , and (). Solve the following POD-based reduced-order FD extrapolating model with fully second-order accuracy:
where and is decided by the classical FD equations (2)-(5) writing as vector form, obtaining the reduced-order solution vectors , , , and , further, obtaining the component forms , , , and (, , , ).
Step 5. Put
If (), then , , , and () are just solutions satisfying accuracy needed. Else, namely, if (), put , , , and (), return to Step 2.
Remark 3 If the classical FD equations (2)-(5) are used to solve NSIBEs, since it includes a large number of degrees of freedom and the truncation error is accumulated in the computational process, it may appear to have no convergence after some computing steps; while if the POD-based reduced-order FD extrapolating model (11)-(13) is used to find the numerical solutions for NSIBEs, since it includes fewer degrees of freedom, it can lessen the truncation error accumulation in the computational process and continuously simulate the development of the fluid flow.