# American Institute of Mathematical Sciences

October  2020, 25(10): 3807-3830. doi: 10.3934/dcdsb.2020229

## Higher-order time-stepping schemes for fluid-structure interaction problems

 1 Computer, Electrical and Mathematical Sciences and Engineering, King Abdullah University of Science and Technology, Thuwal 23955-6900, Saudi Arabia 2 Dipartimento di Matematica "F. Casorati", University of Pavia, Pavia, Italy 3 DICATAM, University of Brescia, Brescia, Italy 4 Technische Universität München (TUM), München, Germany

* Corresponding author: Daniele Boffi

Received  June 2019 Revised  March 2020 Published  July 2020

We consider a recently introduced formulation for fluid-structure interaction problems which makes use of a distributed Lagrange multiplier in the spirit of the fictitious domain method. In this paper we focus on time integration methods of second order based on backward differentiation formulae and on the Crank–Nicolson method. We show the stability properties of the resulting method; numerical tests confirm the theoretical results.

Citation: Daniele Boffi, Lucia Gastaldi, Sebastian Wolf. Higher-order time-stepping schemes for fluid-structure interaction problems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (10) : 3807-3830. doi: 10.3934/dcdsb.2020229
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##### References:
Geometrical configuration of the FSI problem
Sparsity pattern for a matrix arising from Equation (24)
Meshes for the fluid and the structure
The deformed annulus 1
Volume preservation over time
Volume preservation over time for coarser parameters
Volume preservation over time for IFEM method
Mesh parameters
 DOFs $\mathbf{u}_h$ DOFs $p_h$ DOFs $\mathbf{X}_h$ DOFs $\lambda_h$ coarse mesh (M = $8$) $578$ $209$ $306$ $306$ fine mesh (M = $16$) $2,178$ $801$ $1,122$ $1,122$
 DOFs $\mathbf{u}_h$ DOFs $p_h$ DOFs $\mathbf{X}_h$ DOFs $\lambda_h$ coarse mesh (M = $8$) $578$ $209$ $306$ $306$ fine mesh (M = $16$) $2,178$ $801$ $1,122$ $1,122$
Convergence results for the fully implicit scheme on the coarse mesh
 Fluid velocity $\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$ $\Delta t$ $L^2$ error rate $L^2$ error rate $L^2$ error rate $L^2$ error rate $0.05$ $7.63\cdot 10^{-2}$ $2.97\cdot 10^{-2}$ $2.37\cdot 10^{-1}$ $2.42\cdot 10^{-1}$ $0.025$ $4.11\cdot 10^{-2}$ $0.89$ $4.90\cdot 10^{-3}$ $2.60$ $6.24\cdot 10^{-2}$ $1.92$ $6.02\cdot 10^{-2}$ $2.00$ $0.0125$ $2.13\cdot 10^{-2}$ $0.95$ $1.13\cdot 10^{-3}$ $2.11$ $1.21\cdot 10^{-2}$ $2.36$ $1.10\cdot 10^{-2}$ $2.45$ $0.00625$ $1.08\cdot 10^{-2}$ $0.97$ $2.86\cdot 10^{-4}$ $1.98$ $2.03\cdot 10^{-3}$ $2.58$ $9.95\cdot 10^{-4}$ $3.47$ Structure deformation $\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$ $\Delta t$ $L^2$ error rate $L^2$ error rate $L^2$ error rate $L^2$ error rate $0.05$ $1.60\cdot 10^{-3}$ $4.39\cdot 10^{-4}$ $1.43\cdot 10^{-3}$ $2.95\cdot 10^{-4}$ $0.025$ $8.40\cdot 10^{-4}$ $0.93$ $9.75\cdot 10^{-5}$ $2.17$ $7.90\cdot 10^{-4}$ $0.86$ $6.89\cdot 10^{-5}$ $2.10$ $0.0125$ $4.29\cdot 10^{-4}$ $0.97$ $2.53\cdot 10^{-5}$ $1.95$ $4.06\cdot 10^{-4}$ $0.96$ $7.53\cdot 10^{-6}$ $3.19$ $0.00625$ $2.17\cdot 10^{-4}$ $0.98$ $6.41\cdot 10^{-6}$ $1.98$ $2.04\cdot 10^{-4}$ $0.99$ $2.05\cdot 10^{-6}$ $1.88$
 Fluid velocity $\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$ $\Delta t$ $L^2$ error rate $L^2$ error rate $L^2$ error rate $L^2$ error rate $0.05$ $7.63\cdot 10^{-2}$ $2.97\cdot 10^{-2}$ $2.37\cdot 10^{-1}$ $2.42\cdot 10^{-1}$ $0.025$ $4.11\cdot 10^{-2}$ $0.89$ $4.90\cdot 10^{-3}$ $2.60$ $6.24\cdot 10^{-2}$ $1.92$ $6.02\cdot 10^{-2}$ $2.00$ $0.0125$ $2.13\cdot 10^{-2}$ $0.95$ $1.13\cdot 10^{-3}$ $2.11$ $1.21\cdot 10^{-2}$ $2.36$ $1.10\cdot 10^{-2}$ $2.45$ $0.00625$ $1.08\cdot 10^{-2}$ $0.97$ $2.86\cdot 10^{-4}$ $1.98$ $2.03\cdot 10^{-3}$ $2.58$ $9.95\cdot 10^{-4}$ $3.47$ Structure deformation $\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$ $\Delta t$ $L^2$ error rate $L^2$ error rate $L^2$ error rate $L^2$ error rate $0.05$ $1.60\cdot 10^{-3}$ $4.39\cdot 10^{-4}$ $1.43\cdot 10^{-3}$ $2.95\cdot 10^{-4}$ $0.025$ $8.40\cdot 10^{-4}$ $0.93$ $9.75\cdot 10^{-5}$ $2.17$ $7.90\cdot 10^{-4}$ $0.86$ $6.89\cdot 10^{-5}$ $2.10$ $0.0125$ $4.29\cdot 10^{-4}$ $0.97$ $2.53\cdot 10^{-5}$ $1.95$ $4.06\cdot 10^{-4}$ $0.96$ $7.53\cdot 10^{-6}$ $3.19$ $0.00625$ $2.17\cdot 10^{-4}$ $0.98$ $6.41\cdot 10^{-6}$ $1.98$ $2.04\cdot 10^{-4}$ $0.99$ $2.05\cdot 10^{-6}$ $1.88$
Convergence results for the fully implicit scheme on the fine mesh
 Fluid velocity $\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$ $\Delta t$ $L^2$ error rate $L^2$ error rate $L^2$ error rate $L^2$ error rate $0.05$ $9.05\cdot 10^{-2}$ $3.62\cdot 10^{-2}$ $2.28\cdot 10^{-1}$ $2.26\cdot 10^{-1}$ $0.025$ $4.87\cdot 10^{-2}$ $0.89$ $5.05\cdot 10^{-3}$ $2.84$ $6.23\cdot 10^{-2}$ $1.87$ $6.04\cdot 10^{-2}$ $1.91$ $0.0125$ $2.54\cdot 10^{-2}$ $0.94$ $1.20\cdot 10^{-3}$ $2.07$ $2.28\cdot 10^{-2}$ $1.45$ $2.07\cdot 10^{-2}$ $1.54$ $0.00625$ $1.29\cdot 10^{-2}$ $0.98$ $3.53\cdot 10^{-4}$ $1.77$ $5.27\cdot 10^{-3}$ $2.11$ $4.03\cdot 10^{-3}$ $2.36$ Fluid velocity $\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$ $\Delta t$ $L^2$ error rate $L^2$ error rate $L^2$ error rate $L^2$ error rate $0.05$ $1.98\cdot 10^{-3}$ $5.19\cdot 10^{-4}$ $1.65\cdot 10^{-3}$ $4.04\cdot 10^{-4}$ $0.025$ $1.05\cdot 10^{-3}$ $0.92$ $9.79\cdot 10^{-5}$ $2.41$ $9.27\cdot 10^{-4}$ $0.84$ $8.48\cdot 10^{-5}$ $2.25$ $0.0125$ $5.31\cdot 10^{-4}$ $0.99$ $3.13\cdot 10^{-5}$ $1.64$ $4.90\cdot 10^{-4}$ $0.92$ $2.47\cdot 10^{-5}$ $1.78$ $0.00625$ $2.70\cdot 10^{-4}$ $0.98$ $1.35\cdot 10^{-5}$ $1.22$ $2.50\cdot 10^{-4}$ $0.97$ $3.47\cdot 10^{-6}$ $2.83$
 Fluid velocity $\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$ $\Delta t$ $L^2$ error rate $L^2$ error rate $L^2$ error rate $L^2$ error rate $0.05$ $9.05\cdot 10^{-2}$ $3.62\cdot 10^{-2}$ $2.28\cdot 10^{-1}$ $2.26\cdot 10^{-1}$ $0.025$ $4.87\cdot 10^{-2}$ $0.89$ $5.05\cdot 10^{-3}$ $2.84$ $6.23\cdot 10^{-2}$ $1.87$ $6.04\cdot 10^{-2}$ $1.91$ $0.0125$ $2.54\cdot 10^{-2}$ $0.94$ $1.20\cdot 10^{-3}$ $2.07$ $2.28\cdot 10^{-2}$ $1.45$ $2.07\cdot 10^{-2}$ $1.54$ $0.00625$ $1.29\cdot 10^{-2}$ $0.98$ $3.53\cdot 10^{-4}$ $1.77$ $5.27\cdot 10^{-3}$ $2.11$ $4.03\cdot 10^{-3}$ $2.36$ Fluid velocity $\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$ $\Delta t$ $L^2$ error rate $L^2$ error rate $L^2$ error rate $L^2$ error rate $0.05$ $1.98\cdot 10^{-3}$ $5.19\cdot 10^{-4}$ $1.65\cdot 10^{-3}$ $4.04\cdot 10^{-4}$ $0.025$ $1.05\cdot 10^{-3}$ $0.92$ $9.79\cdot 10^{-5}$ $2.41$ $9.27\cdot 10^{-4}$ $0.84$ $8.48\cdot 10^{-5}$ $2.25$ $0.0125$ $5.31\cdot 10^{-4}$ $0.99$ $3.13\cdot 10^{-5}$ $1.64$ $4.90\cdot 10^{-4}$ $0.92$ $2.47\cdot 10^{-5}$ $1.78$ $0.00625$ $2.70\cdot 10^{-4}$ $0.98$ $1.35\cdot 10^{-5}$ $1.22$ $2.50\cdot 10^{-4}$ $0.97$ $3.47\cdot 10^{-6}$ $2.83$
Maximum iterates of the nonlinear solver on the coarse mesh
 $\Delta t$ $\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$ $0.05$ $5$ $5$ $4$ $7$ $0.025$ $4$ $4$ $3$ $4$ $0.0125$ $3$ $3$ $3$ $3$ $0.00625$ $3$ $3$ $2$ $3$
 $\Delta t$ $\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$ $0.05$ $5$ $5$ $4$ $7$ $0.025$ $4$ $4$ $3$ $4$ $0.0125$ $3$ $3$ $3$ $3$ $0.00625$ $3$ $3$ $2$ $3$
Maximum iterates of the nonlinear solver on the fine mesh
 $\Delta t$ $\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$ $0.05$ $10$ $5$ $6$ $6$ $0.025$ $6$ $5$ $5$ $4$ $0.0125$ $6$ $4$ $4$ $4$ $0.00625$ $4$ $4$ $3$ $3$
 $\Delta t$ $\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$ $0.05$ $10$ $5$ $6$ $6$ $0.025$ $6$ $5$ $5$ $4$ $0.0125$ $6$ $4$ $4$ $4$ $0.00625$ $4$ $4$ $3$ $3$
Convergence results for the semi-implicit scheme on the coarse mesh
 Fluid velocity $\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$ $\Delta t$ $L^2$ error rate $L^2$ error rate $L^2$ error rate $L^2$ error rate $0.05$ $7.78\cdot 10^{-2}$ $3.05\cdot 10^{-2}$ $2.49\cdot 10^{-1}$ $2.58\cdot 10^{-1}$ $0.025$ $4.17\cdot 10^{-2}$ $0.90$ $7.89\cdot 10^{-3}$ $1.95$ $6.24\cdot 10^{-2}$ $2.00$ $6.74\cdot 10^{-2}$ $1.94$ $0.0125$ $2.17\cdot 10^{-2}$ $0.95$ $3.14\cdot 10^{-3}$ $1.33$ $1.24\cdot 10^{-2}$ $2.33$ $2.64\cdot 10^{-2}$ $1.35$ $0.00625$ $1.10\cdot 10^{-2}$ $0.97$ $1.29\cdot 10^{-3}$ $1.29$ $2.25\cdot 10^{-3}$ $2.47$ $3.18\cdot 10^{-3}$ $3.06$ Structure deformation $\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$ $\Delta t$ $L^2$ error rate $L^2$ error rate $L^2$ error rate $L^2$ error rate $0.05$ $1.67\cdot 10^{-3}$ $6.79\cdot 10^{-4}$ $1.44\cdot 10^{-3}$ $3.52\cdot 10^{-4}$ $0.025$ $8.65\cdot 10^{-4}$ $0.95$ $2.70\cdot 10^{-4}$ $1.33$ $7.91\cdot 10^{-4}$ $0.86$ $2.60\cdot 10^{-4}$ $0.44$ $0.0125$ $4.36\cdot 10^{-4}$ $0.99$ $1.24\cdot 10^{-4}$ $1.12$ $4.05\cdot 10^{-4}$ $0.97$ $1.53\cdot 10^{-5}$ $4.08$ $0.00625$ $2.17\cdot 10^{-4}$ $1.01$ $5.71\cdot 10^{-5}$ $1.12$ $2.05\cdot 10^{-4}$ $0.98$ $9.43\cdot 10^{-6}$ $0.70$
 Fluid velocity $\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$ $\Delta t$ $L^2$ error rate $L^2$ error rate $L^2$ error rate $L^2$ error rate $0.05$ $7.78\cdot 10^{-2}$ $3.05\cdot 10^{-2}$ $2.49\cdot 10^{-1}$ $2.58\cdot 10^{-1}$ $0.025$ $4.17\cdot 10^{-2}$ $0.90$ $7.89\cdot 10^{-3}$ $1.95$ $6.24\cdot 10^{-2}$ $2.00$ $6.74\cdot 10^{-2}$ $1.94$ $0.0125$ $2.17\cdot 10^{-2}$ $0.95$ $3.14\cdot 10^{-3}$ $1.33$ $1.24\cdot 10^{-2}$ $2.33$ $2.64\cdot 10^{-2}$ $1.35$ $0.00625$ $1.10\cdot 10^{-2}$ $0.97$ $1.29\cdot 10^{-3}$ $1.29$ $2.25\cdot 10^{-3}$ $2.47$ $3.18\cdot 10^{-3}$ $3.06$ Structure deformation $\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$ $\Delta t$ $L^2$ error rate $L^2$ error rate $L^2$ error rate $L^2$ error rate $0.05$ $1.67\cdot 10^{-3}$ $6.79\cdot 10^{-4}$ $1.44\cdot 10^{-3}$ $3.52\cdot 10^{-4}$ $0.025$ $8.65\cdot 10^{-4}$ $0.95$ $2.70\cdot 10^{-4}$ $1.33$ $7.91\cdot 10^{-4}$ $0.86$ $2.60\cdot 10^{-4}$ $0.44$ $0.0125$ $4.36\cdot 10^{-4}$ $0.99$ $1.24\cdot 10^{-4}$ $1.12$ $4.05\cdot 10^{-4}$ $0.97$ $1.53\cdot 10^{-5}$ $4.08$ $0.00625$ $2.17\cdot 10^{-4}$ $1.01$ $5.71\cdot 10^{-5}$ $1.12$ $2.05\cdot 10^{-4}$ $0.98$ $9.43\cdot 10^{-6}$ $0.70$
Convergence results for the semi-implicit scheme on the fine mesh
 Fluid velocity $\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$ $\Delta t$ $L^2$ error rate $L^2$ error rate $L^2$ error rate $L^2$ error rate $0.05$ $9.18\cdot 10^{-2}$ $3.89\cdot 10^{-2}$ $2.36\cdot 10^{-1}$ $2.39\cdot 10^{-1}$ $0.025$ $5.05\cdot 10^{-2}$ $0.86$ $8.59\cdot 10^{-3}$ $2.18$ $7.54\cdot 10^{-2}$ $1.64$ $7.06\cdot 10^{-2}$ $1.76$ $0.0125$ $2.63\cdot 10^{-2}$ $0.94$ $3.32\cdot 10^{-3}$ $1.37$ $4.24\cdot 10^{-2}$ $0.83$ $2.22\cdot 10^{-2}$ $1.67$ $0.00625$ $1.33\cdot 10^{-2}$ $0.98$ $1.40\cdot 10^{-3}$ $1.24$ $2.19\cdot 10^{-2}$ $0.96$ $4.19\cdot 10^{-3}$ $2.40$ Structure deformation $\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$ $\Delta t$ $L^2$ error rate $L^2$ error rate $L^2$ error rate $L^2$ error rate $0.05$ $2.03\cdot 10^{-3}$ $7.86\cdot 10^{-4}$ $1.81\cdot 10^{-3}$ $6.51\cdot 10^{-4}$ $0.025$ $1.06\cdot 10^{-3}$ $0.93$ $3.28\cdot 10^{-4}$ $1.26$ $9.75\cdot 10^{-4}$ $0.89$ $1.31\cdot 10^{-4}$ $2.31$ $0.0125$ $5.34\cdot 10^{-4}$ $1.00$ $1.44\cdot 10^{-4}$ $1.18$ $5.10\cdot 10^{-4}$ $0.93$ $4.82\cdot 10^{-5}$ $1.44$ $0.00625$ $2.69\cdot 10^{-4}$ $0.99$ $6.31\cdot 10^{-5}$ $1.19$ $2.55\cdot 10^{-4}$ $1.00$ $1.29\cdot 10^{-5}$ $1.90$
 Fluid velocity $\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$ $\Delta t$ $L^2$ error rate $L^2$ error rate $L^2$ error rate $L^2$ error rate $0.05$ $9.18\cdot 10^{-2}$ $3.89\cdot 10^{-2}$ $2.36\cdot 10^{-1}$ $2.39\cdot 10^{-1}$ $0.025$ $5.05\cdot 10^{-2}$ $0.86$ $8.59\cdot 10^{-3}$ $2.18$ $7.54\cdot 10^{-2}$ $1.64$ $7.06\cdot 10^{-2}$ $1.76$ $0.0125$ $2.63\cdot 10^{-2}$ $0.94$ $3.32\cdot 10^{-3}$ $1.37$ $4.24\cdot 10^{-2}$ $0.83$ $2.22\cdot 10^{-2}$ $1.67$ $0.00625$ $1.33\cdot 10^{-2}$ $0.98$ $1.40\cdot 10^{-3}$ $1.24$ $2.19\cdot 10^{-2}$ $0.96$ $4.19\cdot 10^{-3}$ $2.40$ Structure deformation $\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$ $\Delta t$ $L^2$ error rate $L^2$ error rate $L^2$ error rate $L^2$ error rate $0.05$ $2.03\cdot 10^{-3}$ $7.86\cdot 10^{-4}$ $1.81\cdot 10^{-3}$ $6.51\cdot 10^{-4}$ $0.025$ $1.06\cdot 10^{-3}$ $0.93$ $3.28\cdot 10^{-4}$ $1.26$ $9.75\cdot 10^{-4}$ $0.89$ $1.31\cdot 10^{-4}$ $2.31$ $0.0125$ $5.34\cdot 10^{-4}$ $1.00$ $1.44\cdot 10^{-4}$ $1.18$ $5.10\cdot 10^{-4}$ $0.93$ $4.82\cdot 10^{-5}$ $1.44$ $0.00625$ $2.69\cdot 10^{-4}$ $0.99$ $6.31\cdot 10^{-5}$ $1.19$ $2.55\cdot 10^{-4}$ $1.00$ $1.29\cdot 10^{-5}$ $1.90$
Maximum residual in the semi-implicit scheme on the coarse mesh
 $\Delta t$ $\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$ $0.05$ $3.83\cdot 10^{-3}$ $3.83\cdot 10^{-3}$ $9.87\cdot 10^{-3}$ $9.64\cdot 10^{-2}$ $0.025$ $2.09\cdot 10^{-3}$ $2.30\cdot 10^{-3}$ $1.24\cdot 10^{-3}$ $2.17\cdot 10^{-2}$ $0.0125$ $7.41\cdot 10^{-4}$ $8.26\cdot 10^{-4}$ $3.62\cdot 10^{-4}$ $7.90\cdot 10^{-3}$ $0.00625$ $2.23\cdot 10^{-4}$ $2.45\cdot 10^{-4}$ $1.08\cdot 10^{-4}$ $9.55\cdot 10^{-4}$
 $\Delta t$ $\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$ $0.05$ $3.83\cdot 10^{-3}$ $3.83\cdot 10^{-3}$ $9.87\cdot 10^{-3}$ $9.64\cdot 10^{-2}$ $0.025$ $2.09\cdot 10^{-3}$ $2.30\cdot 10^{-3}$ $1.24\cdot 10^{-3}$ $2.17\cdot 10^{-2}$ $0.0125$ $7.41\cdot 10^{-4}$ $8.26\cdot 10^{-4}$ $3.62\cdot 10^{-4}$ $7.90\cdot 10^{-3}$ $0.00625$ $2.23\cdot 10^{-4}$ $2.45\cdot 10^{-4}$ $1.08\cdot 10^{-4}$ $9.55\cdot 10^{-4}$
Maximum residual in the semi-implicit scheme on the fine mesh
 $\Delta t$ $\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$ $0.05$ $3.59\cdot 10^{-2}$ $3.59\cdot 10^{-2}$ $3.43\cdot 10^{-2}$ $3.81\cdot 10^{-2}$ $0.025$ $1.03\cdot 10^{-2}$ $1.07\cdot 10^{-2}$ $8.19\cdot 10^{-3}$ $1.02\cdot 10^{-2}$ $0.0125$ $5.28\cdot 10^{-3}$ $5.87\cdot 10^{-3}$ $1.54\cdot 10^{-3}$ $2.27\cdot 10^{-3}$ $0.00625$ $1.46\cdot 10^{-3}$ $1.44\cdot 10^{-3}$ $4.33\cdot 10^{-4}$ $5.08\cdot 10^{-4}$
 $\Delta t$ $\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$ $0.05$ $3.59\cdot 10^{-2}$ $3.59\cdot 10^{-2}$ $3.43\cdot 10^{-2}$ $3.81\cdot 10^{-2}$ $0.025$ $1.03\cdot 10^{-2}$ $1.07\cdot 10^{-2}$ $8.19\cdot 10^{-3}$ $1.02\cdot 10^{-2}$ $0.0125$ $5.28\cdot 10^{-3}$ $5.87\cdot 10^{-3}$ $1.54\cdot 10^{-3}$ $2.27\cdot 10^{-3}$ $0.00625$ $1.46\cdot 10^{-3}$ $1.44\cdot 10^{-3}$ $4.33\cdot 10^{-4}$ $5.08\cdot 10^{-4}$
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