A non-iterative domain decomposition time integrator for linear wave equations

Marlis Hochbruck; Tim Buchholz

arxiv: 2507.19379 · v2 · submitted 2025-07-25 · 🧮 math.NA · cs.NA

A non-iterative domain decomposition time integrator for linear wave equations

Tim Buchholz , Marlis Hochbruck This is my paper

Pith reviewed 2026-05-19 03:01 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords domain decompositionwave equationnon-iterative integratorCrank-Nicolsonfinite elementsmass lumpingparallel time steppingCFL condition

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The pith

A non-iterative domain decomposition integrator for the linear acoustic wave equation achieves second-order accuracy in time and global convergence of order O(h + τ²) under a CFL-type condition on subdomain overlap.

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

The paper introduces a method that divides the computational domain into overlapping subdomains for solving the linear acoustic wave equation. Inside each subdomain an implicit Crank-Nicolson step is taken while a simple local prediction handles the interfaces. This setup allows computations in different subdomains to run in parallel and avoids any iterative solving at each time step. The approach adapts earlier ideas from parabolic problems to hyperbolic wave equations and comes with a proof of second-order accuracy in time together with overall convergence of order O(h + τ²) as long as the overlap width satisfies a certain stability condition. Such a method matters because explicit schemes for waves often force very small time steps while this one can use larger steps without losing accuracy and supports efficient parallel implementation.

Core claim

The authors propose and analyze a non-iterative domain decomposition time integrator for the linear acoustic wave equation. By combining an implicit Crank-Nicolson step on spatial subdomains with a local prediction step at the subdomain interfaces the method enables parallelization across space while advancing sequentially in time without iterations. Using linear finite elements with mass lumping the method is shown to achieve second-order accuracy in time and global convergence of order O(h + τ²) under a CFL-type condition that depends on the overlap width between subdomains.

What carries the argument

The combination of implicit Crank-Nicolson steps on overlapping subdomains with local interface predictions, which allows non-iterative parallel time integration for the wave equation.

Load-bearing premise

The subdomains must overlap by a width large enough to satisfy the CFL-type condition required for stability and the stated convergence rate.

What would settle it

A numerical experiment in which the observed global error fails to decrease as O(h + τ²) when both mesh size h and time step τ are refined while keeping the required overlap width would falsify the convergence claim.

Figures

Figures reproduced from arXiv: 2507.19379 by Marlis Hochbruck, Tim Buchholz.

**Figure 1.** Figure 1: Non-overlapping and overlapping decomposition for ℓ = 3 on an equidistant mesh and for ℓ = 2 on a non-equidistant mesh. 3.2. Crank–Nicolson step on subdomains. The prediction step provides inhomogeneous Dirichlet boundary conditions on all interfaces Γδ i , i = 1, . . . , N. This enables us to apply a Crank–Nicolson step on each subdomain Ωδ i . Using the Crank–Nicolson method on subdomains is considerabl… view at source ↗

**Figure 2.** Figure 2: Maximal stable step sizes τmax of the domain splitting (DSℓ) depending on the overlap parameter ℓ. 10−4 10−3 10−2 10−11 10−8 10−5 10−2 101 τ H1 0 × L 2 error CN DS40 DS16 DS8 DS1 lf O(τ 2 ) [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗

**Figure 3.** Figure 3: Errors of the domain splitting approximation (DSℓ) for the 1d example of Section 9.1 for various numbers of ℓ measured at the final time T = 5.0 against the exact solution (solid) and against the Crank–Nicolson approximation (dashed). For comparison, we also show the error of the Crank–Nicolson (CN) and the leapfrog (lf) methods on the full domain. hmin = 3 · 10−4 and maximum mesh width hmax = 6.9 · 10−4 w… view at source ↗

**Figure 4.** Figure 4: Errors of the domain splitting approximation (DSℓ) with 4 × 4 subdomains and ℓ = 8 for the example from Section 9.2 measured at the final time T = 1.0 against the exact solution (solid) and against the Crank–Nicolson approximation (dashed). For comparison, we also show the error of the Crank–Nicolson (CN) and the leapfrog (lf) methods on the full domain. 10−3 10−2 100 10−2 10−5 10−8 10−10 τ H1 0 × L 2 erro… view at source ↗

**Figure 5.** Figure 5: Errors of the domain splitting approximation against the Crank–Nicolson approximation on different, rectangular subdomain configurations. All runs were made with overlap parameter ℓ = 8. 9.2. Two-dimensional experiments. Let Ω = (0, 1)2 , ξ = 0.5, s = 0.2, and Using (57), we prescribe the exact solution of (1) as u(x, y, t) = u1D(x, t)µξ,s(y) + u1D(y, t)µξ,s(x), where u1D solves the homogeneous one-dimens… view at source ↗

read the original abstract

We propose and analyze a non-iterative domain decomposition integrator for the linear acoustic wave equation. The core idea is to combine an implicit Crank-Nicolson step on spatial subdomains with a local prediction step at the subdomain interfaces. This enables parallelization across space while advancing sequentially in time, without requiring iterations at each time step. The method is similar to the methods from Blum, Lisky and Rannacher (1992) or Dawson and Dupont (1992), which have been designed for parabolic problems. Our approach adapts them to the case of the wave equation in a fully discrete setting, using linear finite elements with mass lumping. Compared to explicit schemes, our method permits significantly larger time steps and retains high accuracy. We prove that the resulting method achieves second-order accuracy in time and global convergence of order $\mathcal{O}(h + \tau^2)$ under a CFL-type condition, which depends on the overlap width between subdomains. We conclude with numerical experiments which 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.

Desk Editor's Note private letter to a colleague

Adapts 1992 parabolic domain decomposition to the wave equation with a non-iterative interface prediction and a claimed O(h + τ²) convergence proof, but the overlap-dependent CFL may limit time-step gains for fixed decompositions.

read the letter

This paper presents a non-iterative domain decomposition time integrator for the linear acoustic wave equation. It uses Crank-Nicolson on overlapping subdomains and adds a local prediction at the interfaces to avoid iterations. The approach takes the 1992 methods for parabolic equations and adjusts them for the wave case in a fully discrete finite element setting with mass lumping. The new part is the adaptation to hyperbolic problems along with the convergence analysis. They show second-order accuracy in time and global error of order O(h + τ²) under a CFL-type condition that involves the overlap width. Numerical experiments are included to support the theory. This setup allows parallel solves across subdomains while advancing in time sequentially, and it claims to permit larger time steps than explicit schemes while keeping accuracy. The analysis seems to hold up on its own terms, with the proof closing under the stated condition. The experiments confirm the rates, which is good to see. One area that could use more attention is how the CFL condition scales with the time step. Since waves propagate at finite speed, the interface prediction likely needs the overlap to be at least on the order of the distance traveled in one step. If the condition ends up requiring the overlap width to increase with τ, then for a fixed decomposition the allowable time step stays bounded. That would make the advantage over standard explicit methods less clear, as those already scale τ with h. The paper should clarify whether fixed overlap allows arbitrarily large τ or not. Overall this is aimed at people developing parallel methods for wave propagation problems in areas like geophysics or acoustics. Readers interested in domain decomposition techniques will find the extension useful. It has enough formal grounding and experimental backing to merit peer review, though revisions might be needed to address the practical implications of the stability condition.

Referee Report

1 major / 2 minor

Summary. The manuscript proposes a non-iterative domain decomposition time integrator for the linear acoustic wave equation. It combines implicit Crank-Nicolson steps on overlapping spatial subdomains with a local prediction step at the interfaces, enabling parallel-in-space, sequential-in-time advancement without iterations per step. The method adapts parabolic DD schemes (Blum-Lisky-Rannacher 1992; Dawson-Dupont 1992) to the hyperbolic case using linear finite elements with mass lumping. The central claims are a proof of second-order temporal accuracy together with global convergence of order O(h + τ²) under a CFL-type condition that depends on the overlap width δ, plus numerical experiments that confirm the theory.

Significance. If the stability and convergence analysis holds, the approach could allow substantially larger time steps than standard explicit schemes for wave propagation while retaining accuracy and enabling spatial parallelism. This would be of practical value for large-scale acoustic simulations provided the overlap requirement does not force δ to grow with τ.

major comments (1)

[stability and convergence analysis] The statement of the CFL-type condition (in the stability and convergence analysis following the method definition): the manuscript asserts that the condition depends on the overlap width δ, yet does not exhibit the explicit form of the restriction (e.g., whether δ ≳ τ or δ ≳ cτ with c>1 is required). Because the local interface prediction must transport information at finite speed for the wave equation, this detail is load-bearing for the claim that the scheme permits significantly larger τ than explicit methods for fixed-overlap decompositions.

minor comments (2)

[introduction] The abstract and introduction cite the 1992 parabolic references but do not clarify which specific technical ingredients (e.g., the prediction step or the mass-lumping treatment) are new versus direct adaptations; a short paragraph contrasting the hyperbolic versus parabolic cases would improve readability.
[numerical experiments] Numerical experiments section: the reported convergence rates are consistent with theory, but the tables or figures do not list the precise ratios δ/τ used in each run, making it harder for the reader to verify that the CFL condition is satisfied independently of the claimed advantage.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for pointing out the need for a more explicit statement of the CFL-type condition. We address this comment below and will update the manuscript accordingly.

read point-by-point responses

Referee: The statement of the CFL-type condition (in the stability and convergence analysis following the method definition): the manuscript asserts that the condition depends on the overlap width δ, yet does not exhibit the explicit form of the restriction (e.g., whether δ ≳ τ or δ ≳ cτ with c>1 is required). Because the local interface prediction must transport information at finite speed for the wave equation, this detail is load-bearing for the claim that the scheme permits significantly larger τ than explicit methods for fixed-overlap decompositions.

Authors: We agree with the referee that an explicit form of the CFL condition would improve the clarity of the stability and convergence analysis. In our proof, the condition is derived to be δ ≥ τ (with the wave speed normalized to 1), ensuring that the interface prediction step can propagate the necessary information across the overlap without loss of accuracy. We will revise the manuscript to state this condition explicitly in the theorem and in the paragraph following the method definition. This explicit form confirms that for a fixed overlap width δ, the allowable time step τ is O(δ), which is independent of the mesh size h and thus significantly larger than the O(h) restriction of standard explicit schemes when h is small. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is a self-contained mathematical proof

full rationale

The paper derives a non-iterative domain decomposition scheme for the linear wave equation by adapting parabolic techniques from external 1992 references, then proves second-order temporal accuracy and O(h + τ²) global convergence via standard finite-element energy estimates under an explicitly stated CFL-type condition on overlap width. No step equates a claimed prediction or result to its own inputs by construction, renames a fitted quantity, or relies on a load-bearing self-citation whose content is unverified. The cited parabolic works are independent external sources; the hyperbolic adaptation and stability analysis are developed directly from the discrete scheme and mass-lumped linear elements without smuggling ansatzes or reducing the theorem to a tautology. The result is therefore self-contained against its stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard finite-element theory and mass lumping but introduces a new non-iterative interface treatment whose stability hinges on an overlap condition.

axioms (1)

domain assumption Subdomains overlap by a width sufficient to satisfy the CFL-type condition for stability and convergence.
The stated convergence order O(h + τ²) is conditional on this overlap requirement.

pith-pipeline@v0.9.0 · 5703 in / 1271 out tokens · 54192 ms · 2026-05-19T03:01:43.691867+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We prove … global convergence of order O(h + τ²) under a CFL-type condition, which depends on the overlap width between subdomains.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

CFL condition (18) τ²‖Lh‖_{Hh←Hh} ≤ 4ℓ²

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages

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Matthew W. Scroggs, Jørgen S. Dokken, Chris N. Richardson, and Garth N. Wells, Construc- tion of arbitrary order finite element degree-of-freedom maps on polygonal and polyhedral cell meshes, ACM Transactions on Mathematical Software 48 (2022), no. 2, 18:1–18:23

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Baratta, Joseph P

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H. Blum, S. Lisky, and R. Rannacher, A domain splitting algorithm for parabolic problems , Computing 49 (1992), no. 1, 11–23. MR1182439

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Brenner and L

Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element meth- ods, Third, Texts in Applied Mathematics, vol. 15, Springer, New York, 2008. MR2373954

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Caffarelli and Michael G

Luis A. Caffarelli and Michael G. Crandall, Distance functions and almost global solu- tions of eikonal equations , Comm. Partial Differential Equations 35 (2010), no. 3, 391–414. MR2748630

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With a foreword by Patrick Joly

Gary Cohen and S´ ebastien Pernet, Finite element and discontinuous Galerkin methods for transient wave equations, Scientific Computation, Springer, Dordrecht, 2017. With a foreword by Patrick Joly. MR3526765

work page 2017

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Dawson and Todd F

Clint N. Dawson and Todd F. Dupont, Explicit/implicit conservative Galerkin domain de- composition procedures for parabolic problems , Math. Comp. 58 (1992), no. 197, 21–34. MR1106964

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, Noniterative domain decomposition for second order hyperbolic problems, Domain de- composition methods in science and engineering (Como, 1992), 1994, pp. 45–52. MR1262604

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49, Springer Nature, 2023

Willy D¨ orfler, Marlis Hochbruck, Jonas K¨ ohler, Andreas Rieder, Roland Schnaubelt, and Christian Wieners, Wave phenomena: Mathematical analysis and numerical approximation , Vol. 49, Springer Nature, 2023

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Dow Drake, Jay Gopalakrishnan, Joachim Sch¨ oberl, and Christoph Wintersteiger, Conver- gence analysis of some tent-based schemes for linear hyperbolic systems , Math. Comp. 91 (2022), no. 334, 699–733. MR4379973

work page 2022

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72, Springer, Cham, [2021] ©2021

Alexandre Ern and Jean-Luc Guermond, Finite elements I—Approximation and interpola- tion, Texts in Applied Mathematics, vol. 72, Springer, Cham, [2021] ©2021. MR4242224

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73, Springer, Cham, [2021] ©2021

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M. J. Gander, L. Halpern, and F. Nataf, Optimal convergence for overlapping and non- overlapping Schwarz waveform relaxation , Eleventh International Conference on Domain Decomposition Methods (London, 1998), 1999, pp. 27–36. MR1827406

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Martin J Gander and Laurence Halpern, Techniques for locally adaptive time stepping devel- oped over the last two decades , Domain decomposition methods in science and engineering xx, 2013, pp. 377–385

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Gander, Laurence Halpern, and Fr´ ed´ eric Nataf, Optimal Schwarz waveform re- laxation for the one dimensional wave equation , SIAM J

Martin J. Gander, Laurence Halpern, and Fr´ ed´ eric Nataf, Optimal Schwarz waveform re- laxation for the one dimensional wave equation , SIAM J. Numer. Anal. 41 (2003), no. 5, 1643–1681. MR2035001

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Gander, Felix Kwok, and Bankim C

Martin J. Gander, Felix Kwok, and Bankim C. Mandal, Dirichlet-Neumann waveform relax- ation methods for parabolic and hyperbolic problems in multiple subdomains , BIT 61 (2021), no. 1, 173–207. MR4235305

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Gander and Hui Zhang, Schwarz methods by domain truncation , Acta Numer

Martin J. Gander and Hui Zhang, Schwarz methods by domain truncation , Acta Numer. 31 (2022), 1–134. MR4436585

work page 2022

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Gopalakrishnan, J

J. Gopalakrishnan, J. Sch¨ oberl, and C. Wintersteiger, Mapped tent pitching schemes for hyperbolic systems, SIAM J. Sci. Comput. 39 (2017), no. 6, B1043–B1063. MR3725284

work page 2017

[20] [20]

Jay Gopalakrishnan, Peter Monk, and Paulina Sep´ ulveda,A tent pitching scheme motivated by Friedrichs theory, Comput. Math. Appl. 70 (2015), no. 5, 1114–1135. MR3378991

work page 2015

[21] [21]

Laurence Halpern and J´ er´ emie Szeftel,Nonlinear nonoverlapping Schwarz waveform relax- ation for semilinear wave propgation, Math. Comp. 78 (2009), no. 266, 865–889. MR2476563

work page 2009

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David Hipp, A unified error analysis for spatial discretizations of wave-type equations with applications to dynamic boundary conditions , Dissertation, 2017

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David Hipp, Marlis Hochbruck, and Christian Stohrer, Unified error analysis for noncon- forming space discretizations of wave-type equations , IMA J. Numer. Anal. 39 (2019), no. 3, 1206–1245. MR3984056

work page 2019

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Holst, Application of domain decomposition and partition of unity methods in physics and geometry, Domain decomposition methods in science and engineering, 2003, pp

M. Holst, Application of domain decomposition and partition of unity methods in physics and geometry, Domain decomposition methods in science and engineering, 2003, pp. 63–78. MR2093735

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Patrick Joly, Variational methods for time-dependent wave propagation problems , Topics in computational wave propagation, 2003, pp. 201–264. MR2032871

work page 2003

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1, 31–36

Qiong-Gui Lin, Yet another approach to solutions of one-dimensional wave equations with inhomogeneous boundary conditions, American Journal of Physics 90 (January 2022), no. 1, 31–36

work page 2022

[27] [27]

Raviart, The use of numerical integration in finite element methods for solving parabolic equations, Topics in numerical analysis (Proc

P.-A. Raviart, The use of numerical integration in finite element methods for solving parabolic equations, Topics in numerical analysis (Proc. Roy. Irish Acad. Conf., University Coll., Dublin, 1972), 1973, pp. 233–264. MR345428

work page 1972

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Scroggs, Jørgen S

Matthew W. Scroggs, Jørgen S. Dokken, Chris N. Richardson, and Garth N. Wells, Construc- tion of arbitrary order finite element degree-of-freedom maps on polygonal and polyhedral cell meshes, ACM Transactions on Mathematical Software 48 (2022), no. 2, 18:1–18:23

work page 2022

[29] [29]

25, Springer-Verlag, Berlin, 1997

Vidar Thom´ ee,Galerkin finite element methods for parabolic problems , Springer Series in Computational Mathematics, vol. 25, Springer-Verlag, Berlin, 1997. MR1479170 Institute for Applied and Numerical Mathematics, Karlsruhe Institute of Technol- ogy, Englerstr. 2, 76131 Karlsruhe, Germany Email address: {tim.buchholz,marlis.hochbruck}@kit.edu

work page 1997