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The Variable Time-stepping DLN-Ensemble Algorithms for Incompressible Navier-Stokes Equations

Wenlong Pei

Abstract

In the report, we propose a family of variable time-stepping ensemble algorithms for solving multiple incompressible Navier-Stokes equations (NSE) at one pass. The one-leg, two-step methods designed by Dahlquist, Liniger, and Nevanlinna (henceforth the DLN method) are non-linearly stable and second-order accurate under arbitrary time grids. We design the family of variable time-stepping DLN-Ensemble algorithms for multiple systems of NSE and prove that its numerical solutions are stable and second-order accurate in velocity under moderate time-step restrictions. Meanwhile, the family of algorithms can be equivalently implemented by a simple refactorization process: adding time filters on the backward Euler ensemble algorithm. In practice, we raise one time adaptive mechanism (based on the local truncation error criterion) for the family of DLN-Ensemble algorithms to balance accuracy and computational costs. Several numerical tests are to support the main conclusions of the report. The constant step test confirms the second-order convergence and time efficiency. The variable step test verifies the stability of the numerical solutions and the time efficiency of the adaptive mechanism.

The Variable Time-stepping DLN-Ensemble Algorithms for Incompressible Navier-Stokes Equations

Abstract

In the report, we propose a family of variable time-stepping ensemble algorithms for solving multiple incompressible Navier-Stokes equations (NSE) at one pass. The one-leg, two-step methods designed by Dahlquist, Liniger, and Nevanlinna (henceforth the DLN method) are non-linearly stable and second-order accurate under arbitrary time grids. We design the family of variable time-stepping DLN-Ensemble algorithms for multiple systems of NSE and prove that its numerical solutions are stable and second-order accurate in velocity under moderate time-step restrictions. Meanwhile, the family of algorithms can be equivalently implemented by a simple refactorization process: adding time filters on the backward Euler ensemble algorithm. In practice, we raise one time adaptive mechanism (based on the local truncation error criterion) for the family of DLN-Ensemble algorithms to balance accuracy and computational costs. Several numerical tests are to support the main conclusions of the report. The constant step test confirms the second-order convergence and time efficiency. The variable step test verifies the stability of the numerical solutions and the time efficiency of the adaptive mechanism.
Paper Structure (19 sections, 13 theorems, 247 equations, 7 figures, 7 tables, 1 algorithm)

This paper contains 19 sections, 13 theorems, 247 equations, 7 figures, 7 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Recent Works on Ensemble Algorithms
  3. Notations and Preliminaries
  4. Algorithm
  5. Numerical Analysis
  6. Stability and Error Analysis in $L^{2}$-norm
  7. Stability and Error Analysis in $H^{1}$-norm
  8. Numerical Analysis for Pressure
  9. Adaptive Implementation
  10. Numerical Tests
  11. Convergence Test
  12. Efficiency Test
  13. Time Adaptive Test
  14. Conclusions
  15. Proof of Lemma \ref{['lemma:DLN-consistency']}
  16. ...and 4 more sections

Key Result

Lemma 2.1

\newlabellemma:b-bound For any $u,v,w \in H^{1}(\Omega)$, If $u,v,w \in X$, then

Figures (7)

  • Figure 3.1: Refactorization Process on (BEFE-Ensemble)-like algorithm
  • Figure 6.1: CPU time for constant DLN-Ensemble algorithm with $\theta = \frac{2}{3},\frac{2}{\sqrt{5}}$
  • Figure 6.2: Time component functions proposed by Lindberg
  • Figure 6.3: All the algorithms have the true pattern of average kinetic energy and maximum kinetic energy. However adaptive algorithms obtain relatively small errors at the end of the simulation since more time steps are assigned in the simulations of extremely stiff parts of true solutions ($t \geq 1.602$).
  • Figure 6.4: The numerical dissipation of adaptive algorithms is relatively large before $t = 1.602$ but grows slower at the end. Meanwhile, all DLN-Ensemble algorithms have similar patterns of kinetic energy dissipation rates. The highly stiff part ($t \geq 1.6$) is hard to simulate since the estimator of LTE $\widehat{T}_{n+1}$ exceeds the required tolerance ${\tt{Tol}} = 1.\rm{e}-4$ frequently and time step size oscillates near the minimum step size $k_{\rm{min}} = 1.\rm{e}-6$ after $t = 1.602$.
  • ...and 2 more figures

Theorems & Definitions (31)

  • Lemma 2.1
  • proof
  • Remark 1
  • Remark 2
  • Definition 4.1
  • Lemma 4.2
  • proof
  • Remark 3
  • Lemma 4.3
  • proof
  • ...and 21 more