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arxiv: 2606.19811 · v1 · pith:7PF4OU7Onew · submitted 2026-06-18 · 🧮 math.NA · cs.NA· math-ph· math.MP

Second order explicit splitting scheme for fluid-poroelastic structure interaction problems

Yifan Wang , Jeonghun Lee , Suncica Canic This is my paper

Pith reviewed 2026-06-26 16:36 UTC · model grok-4.3

classification 🧮 math.NA cs.NAmath-phmath.MP
keywords Stokes-Biot systemfluid-poroelastic interactionexplicit splitting schemepartitioned methodsBDF2 time steppingstability analysisa priori error estimatesRobin reformulation
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The pith

An explicit second-order partitioned scheme for the Stokes-Biot system is stable under a parabolic CFL condition with second-order temporal accuracy.

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

The paper develops a fully discrete explicit splitting scheme for the time-dependent Stokes-Biot problem on fixed domains that solves the Stokes and Biot subproblems independently and in parallel. It combines BDF2 time stepping with second-order Adams-Bashforth extrapolation of interface data through a Robin reformulation. The main contribution is a rigorous stability proof that yields a closed bound under the parabolic CFL condition together with an a priori error estimate showing that errors in fluid velocity, structure velocity, pore pressure, and elastic displacement are controlled by C times (mesh size to the k plus time step squared) for k from 1 to 3 in bulk energy norms. A sympathetic reader cares because the approach delivers provable accuracy for coupled problems without requiring a monolithic solve at each step.

Core claim

The paper establishes that the proposed scheme, which combines BDF2 discretization with second-order Adams-Bashforth extrapolation of the interface data through a Robin reformulation, admits a closed stability bound under a parabolic CFL condition. Using BDF2 energy identities, a sharp decomposition of the extrapolated interface terms, and discrete trace estimates, it further shows that the total errors in fluid velocity, structure velocity, pore pressure, and elastic displacement are bounded by C times the sum of the kth power of the mesh size and the square of the time step, for k from 1 to 3, in bulk energy norms.

What carries the argument

BDF2 energy identities combined with sharp decomposition of extrapolated interface terms and discrete trace estimates, which together yield the closed stability bound for the explicit partitioned scheme.

If this is right

  • The scheme permits independent parallel solution of the fluid and poroelastic subproblems at each time step.
  • Second-order temporal convergence and optimal-order spatial convergence hold in the bulk energy norms for the listed variables.
  • Stability is guaranteed only when the time step obeys the parabolic CFL restriction relative to the mesh size.
  • Numerical tests confirm the rates on manufactured solutions and demonstrate extension to a moving-domain Navier-Stokes case.

Where Pith is reading between the lines

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

  • The parallel subproblem structure could reduce wall-clock time in large-scale applications such as tissue perfusion modeling.
  • Adapting the interface extrapolation and CFL restriction to arbitrary Lagrangian-Eulerian mesh motion would be a natural next step for moving-domain problems.
  • The projection-based error framework may apply directly to other explicit Robin-type couplings in multiphysics systems.

Load-bearing premise

The stability and error analysis requires fixed domains and that the time step satisfies the parabolic CFL condition.

What would settle it

A numerical experiment with manufactured solutions that shows instability or convergence rate below second order in time when the parabolic CFL condition is violated.

read the original abstract

Efficient and provably accurate partitioned methods for fluid-poroelastic structure interaction remain challenging because explicit treatment of the Stokes-Biot interface coupling condition can compromise stability. In this work, we develop and analyze a fully discrete, second-order, explicit splitting scheme for the time-dependent Stokes-Biot problem on fixed domains. The method combines BDF2 time stepping with second-order Adams-Bashforth extrapolation of interface data through a Robin reformulation, yielding a partitioned algorithm in which the Stokes and Biot subproblems are solved independently and in parallel at each time step. The main analytical contribution is a rigorous stability and error analysis for this second-order explicit coupling strategy. Using BDF2 energy identities, a sharp decomposition of the extrapolated interface terms, and discrete trace estimates, we prove a closed stability bound under a parabolic CFL condition. We then derive an a priori error estimate through a projection-based framework using a Fortin projection for the fluid variables and Ritz-type projections for the poroelastic variables. The analysis identifies consistency defects from BDF2 time discretization, Adams-Bashforth interface extrapolation, and the projected kinematic relation. It shows that the total errors in fluid velocity, structure velocity, pore pressure, and elastic displacement are bounded by C times the sum of the kth power of the mesh size and the square of the time step, for k from 1 to 3, in bulk energy norms. Numerical experiments with manufactured solutions confirm second-order temporal convergence and optimal-order spatial convergence. We also include a moving-domain example with Navier-Stokes fluid flow, demonstrating applicability beyond the fixed-domain Stokes-Biot setting analyzed.

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

1 major / 2 minor

Summary. The paper develops and analyzes a fully discrete second-order explicit splitting scheme for the time-dependent Stokes-Biot problem on fixed domains. It combines BDF2 time discretization with second-order Adams-Bashforth extrapolation of interface data via a Robin reformulation, allowing independent parallel solves of the Stokes and Biot subproblems. Using BDF2 energy identities, sharp decomposition of extrapolated interface terms, and discrete trace estimates, the authors prove a closed stability bound under a parabolic CFL condition. An a priori error analysis via Fortin projection for fluid variables and Ritz projections for poroelastic variables yields bounds of order O(h^k + Δt²) for k=1,2,3 in bulk energy norms for fluid velocity, structure velocity, pore pressure, and elastic displacement. Numerical experiments on manufactured solutions confirm the rates, with an additional moving-domain Navier-Stokes example shown numerically.

Significance. If the stability and error results hold, the work provides a valuable contribution to partitioned methods for fluid-poroelastic interaction by delivering a provably stable explicit scheme with optimal-order convergence, which is useful for efficiency in applications such as biomechanics. The rigorous use of energy methods, projection operators, and explicit identification of consistency defects from time discretization and interface extrapolation strengthens the analysis in the numerical PDE literature. The explicit scoping to fixed domains with CFL dependence stated upfront avoids overclaiming.

major comments (1)
  1. [§3] The error analysis claims O(h^k + Δt²) bounds for k ranging from 1 to 3 depending on the projection choice, but the specific mapping of k to each variable (fluid velocity vs. pore pressure) and the precise order of the consistency defect from the projected kinematic relation are not load-bearing if the overall second-order temporal claim is preserved; however, §3 (projection framework) would benefit from an explicit table or remark clarifying which k applies to which norm to avoid ambiguity in the central error theorem.
minor comments (2)
  1. [Introduction / Theorem 3.1] The abstract states the CFL condition is required for the closed stability bound, but the precise form of the parabolic CFL (e.g., Δt ≤ C h² or similar) should be stated explicitly in the introduction or stability theorem statement for immediate readability.
  2. [§5] Numerical section: the moving-domain Navier-Stokes example is presented only numerically without analysis; a brief remark on why the fixed-domain analysis does not directly extend would clarify the scope.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment and the helpful suggestion for improving clarity in the error analysis section. We address the single major comment below.

read point-by-point responses

  1. Referee: [§3] The error analysis claims O(h^k + Δt²) bounds for k ranging from 1 to 3 depending on the projection choice, but the specific mapping of k to each variable (fluid velocity vs. pore pressure) and the precise order of the consistency defect from the projected kinematic relation are not load-bearing if the overall second-order temporal claim is preserved; however, §3 (projection framework) would benefit from an explicit table or remark clarifying which k applies to which norm to avoid ambiguity in the central error theorem.

    Authors: We agree that an explicit mapping would reduce potential ambiguity for readers. In the revised manuscript we will insert a short remark (or compact table) immediately after the statement of the projection operators in §3, listing the approximation order k chosen for each variable (Fortin projection for fluid velocity/pressure, Ritz projections for structure displacement/velocity and pore pressure) together with the resulting contribution to the consistency defect in the kinematic interface relation. This addition preserves all existing proofs and does not alter the O(h^k + Δt²) statement for k=1,2,3. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via standard analysis

full rationale

The paper's stability bound (under parabolic CFL) and a priori error estimate O(h^k + Δt²) for k=1..3 are obtained from BDF2 energy identities, sharp decomposition of extrapolated interface terms, discrete trace estimates, and a projection framework (Fortin for fluid, Ritz for poroelastic). These steps are standard discrete energy methods and projection arguments for partitioned schemes on fixed domains; they do not reduce by construction to fitted inputs, self-definitions, or load-bearing self-citations. The moving-domain Navier-Stokes case is presented only numerically. The derivation chain is independent of its target result and remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Analysis rests on standard numerical analysis tools (BDF2 energy identities, discrete trace inequalities, Fortin and Ritz projections) that are invoked without new postulates; no free parameters or invented entities are described.

axioms (2)
  • standard math BDF2 energy identities hold for the discrete scheme
    Invoked to obtain the closed stability bound
  • standard math Discrete trace estimates and projection properties for Fortin and Ritz operators
    Used to control interface terms and consistency defects

pith-pipeline@v0.9.1-grok · 5833 in / 1464 out tokens · 37583 ms · 2026-06-26T16:36:22.853261+00:00 · methodology

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