# 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  October 2020 Early access  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
##### References:

show all references

##### 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
 $\Delta t$ $\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$ $0.05$ $3.59\cdot 10^{-2}$ $3.59\cdot 10^{-2}$ $3.43\cdot 10^{-2}$ $3.81\cdot 10^{-2}$ $0.025$ $1.03\cdot 10^{-2}$ $1.07\cdot 10^{-2}$ $8.19\cdot 10^{-3}$ $1.02\cdot 10^{-2}$ $0.0125$ $5.28\cdot 10^{-3}$ $5.87\cdot 10^{-3}$ $1.54\cdot 10^{-3}$ $2.27\cdot 10^{-3}$ $0.00625$ $1.46\cdot 10^{-3}$ $1.44\cdot 10^{-3}$ $4.33\cdot 10^{-4}$ $5.08\cdot 10^{-4}$
 $\Delta t$ $\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$ $0.05$ $3.59\cdot 10^{-2}$ $3.59\cdot 10^{-2}$ $3.43\cdot 10^{-2}$ $3.81\cdot 10^{-2}$ $0.025$ $1.03\cdot 10^{-2}$ $1.07\cdot 10^{-2}$ $8.19\cdot 10^{-3}$ $1.02\cdot 10^{-2}$ $0.0125$ $5.28\cdot 10^{-3}$ $5.87\cdot 10^{-3}$ $1.54\cdot 10^{-3}$ $2.27\cdot 10^{-3}$ $0.00625$ $1.46\cdot 10^{-3}$ $1.44\cdot 10^{-3}$ $4.33\cdot 10^{-4}$ $5.08\cdot 10^{-4}$
 [1] Daniele Boffi, Lucia Gastaldi. Discrete models for fluid-structure interactions: The finite element Immersed Boundary Method. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 89-107. doi: 10.3934/dcdss.2016.9.89 [2] Grégoire Allaire, Alessandro Ferriero. Homogenization and long time asymptotic of a fluid-structure interaction problem. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 199-220. doi: 10.3934/dcdsb.2008.9.199 [3] George Avalos, Roberto Triggiani. Fluid-structure interaction with and without internal dissipation of the structure: A contrast study in stability. Evolution Equations & Control Theory, 2013, 2 (4) : 563-598. doi: 10.3934/eect.2013.2.563 [4] Salim Meddahi, David Mora. Nonconforming mixed finite element approximation of a fluid-structure interaction spectral problem. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 269-287. doi: 10.3934/dcdss.2016.9.269 [5] Qiang Du, M. D. Gunzburger, L. S. Hou, J. Lee. Analysis of a linear fluid-structure interaction problem. Discrete & Continuous Dynamical Systems, 2003, 9 (3) : 633-650. doi: 10.3934/dcds.2003.9.633 [6] Serge Nicaise, Cristina Pignotti. Asymptotic analysis of a simple model of fluid-structure interaction. Networks & Heterogeneous Media, 2008, 3 (4) : 787-813. doi: 10.3934/nhm.2008.3.787 [7] Igor Kukavica, Amjad Tuffaha. Solutions to a fluid-structure interaction free boundary problem. Discrete & Continuous Dynamical Systems, 2012, 32 (4) : 1355-1389. doi: 10.3934/dcds.2012.32.1355 [8] Qingguang Guan, Max Gunzburger. Stability and convergence of time-stepping methods for a nonlocal model for diffusion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1315-1335. doi: 10.3934/dcdsb.2015.20.1315 [9] Oualid Kafi, Nader El Khatib, Jorge Tiago, Adélia Sequeira. Numerical simulations of a 3D fluid-structure interaction model for blood flow in an atherosclerotic artery. Mathematical Biosciences & Engineering, 2017, 14 (1) : 179-193. doi: 10.3934/mbe.2017012 [10] Andro Mikelić, Giovanna Guidoboni, Sunčica Čanić. Fluid-structure interaction in a pre-stressed tube with thick elastic walls I: the stationary Stokes problem. Networks & Heterogeneous Media, 2007, 2 (3) : 397-423. doi: 10.3934/nhm.2007.2.397 [11] George Avalos, Roberto Triggiani. Uniform stabilization of a coupled PDE system arising in fluid-structure interaction with boundary dissipation at the interface. Discrete & Continuous Dynamical Systems, 2008, 22 (4) : 817-833. doi: 10.3934/dcds.2008.22.817 [12] Pavel Eichler, Radek Fučík, Robert Straka. Computational study of immersed boundary - lattice Boltzmann method for fluid-structure interaction. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 819-833. doi: 10.3934/dcdss.2020349 [13] Martina Bukač, Sunčica Čanić. Longitudinal displacement in viscoelastic arteries: A novel fluid-structure interaction computational model, and experimental validation. Mathematical Biosciences & Engineering, 2013, 10 (2) : 295-318. doi: 10.3934/mbe.2013.10.295 [14] George Avalos, Thomas J. Clark. A mixed variational formulation for the wellposedness and numerical approximation of a PDE model arising in a 3-D fluid-structure interaction. Evolution Equations & Control Theory, 2014, 3 (4) : 557-578. doi: 10.3934/eect.2014.3.557 [15] Mehdi Badra, Takéo Takahashi. Feedback boundary stabilization of 2d fluid-structure interaction systems. Discrete & Continuous Dynamical Systems, 2017, 37 (5) : 2315-2373. doi: 10.3934/dcds.2017102 [16] Henry Jacobs, Joris Vankerschaver. Fluid-structure interaction in the Lagrange-Poincaré formalism: The Navier-Stokes and inviscid regimes. Journal of Geometric Mechanics, 2014, 6 (1) : 39-66. doi: 10.3934/jgm.2014.6.39 [17] Sören Bartels, Jakob Keck. Adaptive time stepping in elastoplasticity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 71-88. doi: 10.3934/dcdss.2020323 [18] Fredrik Hellman, Patrick Henning, Axel Målqvist. Multiscale mixed finite elements. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1269-1298. doi: 10.3934/dcdss.2016051 [19] George Avalos, Roberto Triggiani. Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 417-447. doi: 10.3934/dcdss.2009.2.417 [20] Andrea L. Bertozzi, Ning Ju, Hsiang-Wei Lu. A biharmonic-modified forward time stepping method for fourth order nonlinear diffusion equations. Discrete & Continuous Dynamical Systems, 2011, 29 (4) : 1367-1391. doi: 10.3934/dcds.2011.29.1367

2020 Impact Factor: 1.327