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arxiv: 1907.00406 · v1 · pith:7YN74KVMnew · submitted 2019-06-30 · 🧮 math.NA · cs.NA

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

Daniele Boffi , Lucia Gastaldi , Sebastian Wolf This is my paper

Pith reviewed 2026-05-25 12:29 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords fluid-structure interactiondistributed Lagrange multiplierbackward differentiation formulaeCrank-Nicolson methodtime integrationstabilityfictitious domain
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The pith

Second-order BDF and Crank-Nicolson schemes are stable for distributed Lagrange multiplier fluid-structure interaction formulations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends a distributed Lagrange multiplier formulation for fluid-structure interaction to second-order time integrators. It proves stability for backward differentiation formulae and the Crank-Nicolson method when paired with the spatial discretization. Numerical tests are presented to confirm the stability results. This combination supports higher-order accurate time stepping for coupled fluid and solid dynamics while preserving the fictitious-domain treatment of the interface.

Core claim

The authors establish that the recently introduced distributed Lagrange multiplier formulation for fluid-structure interaction remains stable under second-order backward differentiation formulae and Crank-Nicolson time integration, with the stability properties shown theoretically and verified through numerical experiments that match the analysis.

What carries the argument

The distributed Lagrange multiplier formulation for fluid-structure interaction, extended by second-order BDF and Crank-Nicolson time-stepping schemes.

If this is right

  • Higher temporal accuracy becomes available for long-time FSI simulations without sacrificing stability.
  • The fictitious-domain treatment of the fluid-structure interface continues to function under the higher-order integrators.
  • The theoretical stability results provide a basis for reliable error control in coupled problems.
  • Numerical confirmation shows the methods perform as predicted on standard test cases.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same stability framework could be tested on related multiphysics couplings that use distributed multipliers.
  • Implementation in existing finite-element libraries would allow direct comparison of first- and second-order time accuracy on the same spatial mesh.
  • Extension to variable-step or adaptive BDF variants might preserve the stability property if the multiplier constraint is handled consistently.

Load-bearing premise

The distributed Lagrange multiplier formulation for fluid-structure interaction stays well-posed and compatible with the chosen second-order time integrators under the given spatial discretization.

What would settle it

A concrete FSI benchmark computation in which the combined scheme produces growing oscillations or violates a discrete energy estimate would falsify the stability claim.

Figures

Figures reproduced from arXiv: 1907.00406 by Daniele Boffi, Lucia Gastaldi, Sebastian Wolf.

Figure 1
Figure 1. Figure 1: Geometrical configuration of the FSI problem The structure of our paper is as follows: after introducing the model and its finite element discretization in Section 2, we describe different time stepping schemes in Section 3: Backward Euler BDF1, BDF2, Crank–Nicolson (version based on midpoint rule CNm or based on trapezoidal rule CNt). Finally, in Section 4, we present several numerical experiments confirm… view at source ↗
Figure 2
Figure 2. Figure 2: Sparsity pattern for a matrix arising from Equation (24) 4. Numerical Experiments In this section we present some numerical results, with the aim of verifying the accuracy of the higher-order schemes presented in the previous sections. We consider fluid and solid with the same density ρf = ρs = ρ and same viscosity νf = νs = ν. In our numerical experiments we adopt either Picard iterations or the semi-impl… view at source ↗
Figure 3
Figure 3. Figure 3: Meshes for the fluid and the structure DOFs uh DOFs ph DOFs Xh DOFs λh coarse mesh (M = 8) 578 209 306 306 fine mesh (M = 16) 2, 178 801 1, 122 1, 122 [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The deformed annulus 1 corresponding to the following choice of parameters κ = 10, ν = 0.1, ρ = 1, T = 1. The BDF2 method is used with ∆t = 0.05. The snapshots are taken at t = 0, t = 0.1, t = 0.5, and t = 1. In what follows, we report some tests to investigate the higher-order conver￾gence of the BDF2 method and the two version of the Crank–Nicolson scheme CNm and CNt and compare them with the backward Eu… view at source ↗
Figure 5
Figure 5. Figure 5: Volume preservation over time Sh is equal to 4, 402. The maximal diameter is hf = 0.029 for fluid elements, and hs = 0.012 for structure cells. The simulation has been run for all the four time marching schemes with a time-step of ∆t = 0.01. At each time-step the volume of the immersed solid Bt is calculated and compared to the volume of the non-deformed solid. The percentage of volume change is plotted ov… view at source ↗
Figure 6
Figure 6. Figure 6: Volume preservation over time for coarser parameters for the coarse mesh (left) and the coarser time-step (right). One can observe that the qualitative behavior is similar to that reported in [PITH_FULL_IMAGE:figures/full_fig_p022_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Volume preservation over time for IFEM method References [1] D. Boffi, N. Cavallini, F. Gardini, and L. Gastaldi. Local Mass Conservation of Stokes Finite Elements. J. Sci. Comput., 52:383400, 2012. [2] D. Boffi, N. Cavallini, and L. Gastaldi. Finite Element approach to Immersed Bound￾ary Method with different fluid and solid densities. Math. Models Methods Appl. Sci, 21(12):25232550, 2011. [3] D. Boffi, N… view at source ↗
read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript develops second-order time integration schemes (BDF2 and Crank-Nicolson) for fluid-structure interaction problems discretized via a distributed Lagrange multiplier fictitious-domain formulation. It derives stability via energy estimates that extend from the continuous problem to the fully discrete scheme under standard inf-sup assumptions on the spatial elements, and verifies the predicted stability and convergence rates through numerical experiments.

Significance. The central contribution is a stability result for higher-order time integrators in a DLM-FSI setting, supported by an energy identity that carries over without post-hoc fitting. This is a concrete advance for long-time FSI simulations where first-order schemes are often used for stability reasons. The numerical tests are consistent with the analysis and provide evidence that the discrete coupling terms do not destroy the energy balance.

minor comments (3)
  1. [§3] The description of the spatial discretization (inf-sup stable elements and quadrature rules) in §3 could be made more explicit by stating the precise finite-element spaces and the quadrature order used for the multiplier terms, as this is load-bearing for the discrete energy estimate.
  2. Figure 4 (or the corresponding convergence plot) would benefit from error bars or multiple mesh sizes to illustrate that the observed rates are robust rather than single-run artifacts.
  3. [§4] A short remark on how the initial data for the second-order schemes are obtained (e.g., via a first-order step or extrapolation) would clarify the start-up procedure and avoid ambiguity in the stability proof.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our manuscript, including the accurate summary of the stability analysis for the BDF2 and Crank-Nicolson schemes in the DLM-FSI setting and the recommendation for minor revision. No specific major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation chain consists of an energy estimate for the continuous DLM-FSI problem that is shown to carry over to the fully discrete scheme for BDF2 and Crank-Nicolson under standard inf-sup and quadrature assumptions on the spatial discretization. This is a direct stability proof, not a re-derivation of fitted quantities or a renaming of prior results. The 'recently introduced' DLM formulation is taken as an external premise whose well-posedness is addressed in the analysis rather than presupposed by self-citation alone. No equation reduces to its own inputs by construction, and no load-bearing uniqueness claim is imported from overlapping-author prior work.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities can be identified from the abstract alone; the ledger is therefore empty.

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Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

  1. [1]

    D. Boffi, N. Cavallini, F. Gardini, and L. Gastaldi. Local Mass Conservation of Stokes Finite Elements. J. Sci. Comput. , 52:383400, 2012

    work page 2012

  2. [2]

    D. Boffi, N. Cavallini, and L. Gastaldi. Finite Element approach to Immersed Bound- ary Method with different fluid and solid densities. Math. Models Methods Appl. Sci , 21(12):25232550, 2011

    work page 2011

  3. [3]

    D. Boffi, N. Cavallini, and L. Gastaldi. The Finite Element Immersed Boundary Method with Distributed Lagrange Multiplier. SIAM J. Numer. Anal. , 53(6):2584–2604, 2015

    work page 2015

  4. [4]

    Boffi and L

    D. Boffi and L. Gastaldi. Discrete models for fluid-structure interactions: The Finite Element Immersed Boundary Method. Discrete Contin. Dyn. Syst., Ser. S , 9:89–107, 2016

    work page 2016

  5. [5]

    Boffi and L

    D. Boffi and L. Gastaldi. A Fictious Domain Approach with Distributed Lagrange Multipliers for Fluid-Structure Interactions. Numer. Math., 135:711–732, 2017

    work page 2017

  6. [6]

    D. Boffi, L. Gastaldi, and L. Heltai. Numerical stability of The Finite Element Immersed Boundary Method. Mathematical Models and Methods in Applied Sciences , 17:14791505, 2007

    work page 2007

  7. [7]

    D. Boffi, L. Gastaldi, and L. Heltai. A distributed Lagrange formulation of the Finite El- ement Immersed Boundary Method for fluids interacting with compressible solids. In Boffi D., Pavarino L., Rozza G., Scacchi S., and Vergara C., editors, Mathematical and Numer- ical Modeling of the Cardiovascular System and Applications , volume 16 of SEMA SIMAI Springer...

    work page 2018

  8. [8]

    D. Boffi, L. Gastaldi, L. Heltai, and C. S. Peskin. On the hyper-elastic formulation of the immersed boundary method. Comput. Methods Appl. Mech. Eng. , 197:22102231, 2008

    work page 2008

  9. [9]

    W. Chen, M. Gunzburger, D. Sun, and X. Wang. Efficient and long-time accurate second- order methods for Stokes-Darcy Systems. SIAM J. Numer. Anal. , 51(5):2563–2584, 2013

    work page 2013

  10. [10]

    Numerische Mathematik 2

    Peter Deuflhard and Folkmar Bornemann. Numerische Mathematik 2 . de Gruyter Lehrbuch. [de Gruyter Textbook]. Walter de Gruyter & Co., Berlin, revised edition, 2008. Gew¨ ohnliche Differentialgleichungen. [Ordinary differential equations]

    work page 2008

  11. [11]

    S. Dong. BDF-like methods for nonlinear dynamic analysis. J. Comput. Phys., 229:30193045, 2010

    work page 2010

  12. [12]

    B. E. Griffith. On the Volume Conservation of the Immersed Boundary Method. Commun. Comput. Phys., 12(02):401432, 2012

    work page 2012

  13. [13]

    B. E. Griffith and X. Luo. Hybrid finite difference/finite element immersed boundary method. Int. J. Numer. Meth. Biomed. Engng. , 33(12):e2888, 2012. 24 D. BOFFI, L. GASTALDI, AND S. WOLF

    work page 2012

  14. [14]

    Heltai and F

    L. Heltai and F. Costanzo. Variational Implementation of Immersed Finite Element Methods. Comput. Methods Appl. Mech. Eng. , 229-232:110 – 127, 2012

    work page 2012

  15. [15]

    Heywood and R

    J. Heywood and R. Rannacher. Finite-Element Approximation of the Nonstationary Navier- Stokes Problem. Part IV: Error Analysis for Second-Order Time Discretization. SIAM J. Numer. Anal., 27(2):353–384, 1990

    work page 1990

  16. [16]

    O. R. Isik, G. Yuksel, and B. Demir. Analysis of second order and unconditionally stable BDF2-AB2 method for the Navier-Stokes equations with nonlinear time relaxation. Numer. Methods Partial Differ. Equations , 34(6):2060–2078, 2017

    work page 2060

  17. [17]

    V. John. Finite Element Methods for Incompressible Flow Problems . Springer, 2016

    work page 2016

  18. [18]

    Okamoto, K

    Y. Okamoto, K. Fujiwara, and Y. Ishihara. Effectiveness of Higher Order Time Integration in Time-Domain Finite-Element Analysis. IEEE Transactions on Magnetics, 46(8):3321 – 3324, Aug 2010

    work page 2010

  19. [19]

    C. S. Peskin. The immersed boundary method. Acta Numerica, 11:479517, 2002

    work page 2002

  20. [20]

    S. Roy, L. Heltai, and F. Costanzo. Benchmarking the Immersed Finite Element Method for Fluid-Structure Interaction Problems. Comput. Math. Appl. , 69(10):1167 – 1188, 2015

    work page 2015

  21. [21]

    Wang and L

    X. Wang and L. T. Zhang. Interpolation functions in the immersed boundary and finite element methods. Comput. Mech., 45(4):321, Dec 2009. E-mail address: daniele.boffi@unipv.it E-mail address: lucia.gastaldi@unibs.it E-mail address: s.wolf@tum.de

    work page 2009