Pei, Wenlong
(2022)
The Search for Time Accuracy: A Variable Time-stepping Algorithm For Computational Fluid Dynamics.
Doctoral Dissertation, University of Pittsburgh.
(Unpublished)
Abstract
Dahlquist, Liniger, and Nevanlinna proposed a two-step time-stepping scheme for systems of ordinary differential equations (ODEs) in 1983. The little-explored variable time-stepping scheme has advantages in numerical simulations for its fine properties such as unconditional $G$-stability and second-order accuracy. However, this numerical scheme is always avoided for time discretization due to its complex form.
To solve this issue, we simplify its implementation through time filters (pre-filter and post-filter) on a certain first-order implicit method. By adding only a few lines
of code, accuracy will be improved while stability is not sacrificed.
G-stability of the scheme for systems of ODEs corresponds to unconditional, long-time energy stability when applied to flow models. The combination of G-stability and consistency provides the preliminaries for error analysis.
We analyze the method of Dahlquist, Liniger, and Nevanlinna (DLN) as a variable step, time discretization of the unsteady Stokes/Darcy model, and the Navier-Stokes equations. We prove that the kinetic energy is bounded for variable time-steps, show that the method is second-order accurate, characterize its numerical dissipation and prove error estimates.
Moreover, the adaptivity algorithm for this variable time-stepping scheme, highly reducing computation cost as well as keeping time accuracy, has been applied to systems of ODEs and flow models. The local truncation error criterion for adapting time steps, with corresponding error estimators, is known in ODE problems. Many methods of error estimation are possible; herein we focus on ones that involve minimal extra storage and computations. First, we extend a classic and highly efficient idea of Gear from the trapezoid rule to the DLN method. Second, we consider a recent refactorization of the DLN method which eases the implementation of DLN in legacy codes. We show that this refactorization provides methods for effective error estimation, at no extra cost.
For fluid models, the minimum numerical dissipation criterion is used for adjusting the time steps, as the estimator of the local truncation error would be more complicated. The 2D offset circles problem by this algorithm is used to confirm the stability of the approximate solutions.
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Details
| Item Type: |
University of Pittsburgh ETD
|
| Status: |
Unpublished |
| Creators/Authors: |
|
| ETD Committee: |
|
| Date: |
12 October 2022 |
| Date Type: |
Publication |
| Defense Date: |
13 July 2022 |
| Approval Date: |
12 October 2022 |
| Submission Date: |
28 July 2022 |
| Access Restriction: |
No restriction; Release the ETD for access worldwide immediately. |
| Number of Pages: |
125 |
| Institution: |
University of Pittsburgh |
| Schools and Programs: |
Dietrich School of Arts and Sciences > Mathematics |
| Degree: |
PhD - Doctor of Philosophy |
| Thesis Type: |
Doctoral Dissertation |
| Refereed: |
Yes |
| Uncontrolled Keywords: |
computational fluid dynamics, variable time-stepping,
G-stability, second-order convergence,
time adaptivity |
| Date Deposited: |
12 Oct 2022 16:02 |
| Last Modified: |
12 Oct 2022 16:02 |
| URI: |
http://d-scholarship.pitt.edu/id/eprint/43408 |
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