PaperlayerView exact revision on arXiv

Source-linked AI summary

Dirichlet--Neumann waveform relaxation for heterogeneous heat equations: fully discrete $l^2$ analysis

Philipp Birken, Niklas Kotarsky

arXiv:2610.01475v1math.NAcs.NA

TL;DR

The paper addresses limited understanding of fully discrete DNWR error estimates, particularly the effects of unequal spatial grids. It derives finite-time l2 estimates with an exponentially weighted Fourier technique; large and small CFL numbers correspond to different grid-dependence behavior, and numerical experiments validate the estimates across a range of discretizations.

  • Problem

    The effects of spatial mesh differences and the relation between continuous, semidiscrete, and fully discrete DNWR iterations are not fully understood, despite their relevance to coarse and mismatched grids in coupled applications.

    Source: §1 · p. 2

  • Method

    The paper derives finite-time l2 error estimates for fully discrete DNWR on coupled heat equations using an exponentially weighted Fourier technique.

    Source: §1 · p. 3, §3 · p. 6, §6 · p. 15

  • Results

    For large CFL numbers, the estimate behaves like the time-discrete counterpart and is independent of ∆x1 and ∆x2; for small CFL numbers, convergence is expected to depend on ∆x1/∆x2, and experiments validate the estimates across varied discretizations.

    Source: §4.1 · p. 10, §6 · p. 16

  • Takeaways & Limitations

    The estimates characterize how CFL numbers and spatial mesh ratios relate to fully discrete DNWR convergence.

    Source: §6 · pp. 15–16

    The transform-existence lemma assumes a square-summable sequence that is zero for negative time indices.

    Source: §3 · p. 6

Abstract

from arXiv · show

We consider two coupled linear heat equations on different spatial domains that interact through a lower dimensional interface. This models conjugate heat transfer. The problem is solved using Dirichlet--Neumann waveform relaxation. This allows the subproblems to be solved using separate codes, a so called partitioned approach. Our overall goal is to develop more efficient partitioned methods, and to this end, we want reliable error estimates. Here, we use an exponentially weighted Fourier technique to derive new error estimates in $l^2$ for finite time $T$ in the fully discrete setting. These describe both linear and superlinear behavior. We show that the fully discrete estimate is close to a previously obtained time discrete estimate and independent of $\Delta x_1$ and $\Delta x_2$ when the CFL number in the subsolvers is large. We also show that the convergence behaviour depends on the ratio $\Delta x_1/\Delta x_2$ when the CFL number is small. Our numerical experiments show that the fully discrete estimate is accurate across a wide range of grid sizes $\Delta x_1, \Delta x_2$, time step sizes $\Delta t$ and $T$.

1. Introduction

The paper addresses gaps in fully discrete convergence estimates for Dirichlet–Neumann waveform relaxation (DNWR), especially the effects of unequal spatial meshes and the relation between discrete and continuous iterations. It derives finite-time l2 estimates and examines how convergence depends on the subsolvers’ CFL numbers and grid-size ratio.

1. Introduction

  • DNWR supports partitioned solution with separate subsolvers, while suitable relaxation parameters can yield superlinear convergence on finite time intervals.Waveform relaxation alternates solutions of coupled subsystems using data from the other subsystems.Source: §1 · pp. 1–2
  • The unresolved mesh and discretization questions matter in climate science and thermal FSI, where coarse or different grid sizes are commonly used across coupled domains.The paper frames these issues in the context of conjugate heat transfer, including ocean–atmosphere interaction and engineering applications.Source: §1 · pp. 1–2
  • Prior work includes continuous and time-discrete error estimates using exponentially weighted Fourier techniques, including superlinear estimates for a material-dependent relaxation parameter.Fully discrete and space-discrete analyses also used Fourier or z-transforms to obtain linear estimates.Source: §1 · pp. 2–3
  • Existing fully discrete analyses gave linear estimates but did not explain the effect of different grid sizes; a one-step study found CFL- and grid-ratio-dependent convergence.The one-step result found convergence proportional to ∆x1/∆x2 for small CFL numbers, while its extension to multiple time steps remained unclear.Source: §1 · p. 3
  • The paper derives fully discrete l2 error estimates for finite T and finds that large CFL numbers make the estimate close to its time-discrete counterpart and independent of grid sizes, while small CFL numbers bring grid-ratio dependence.The analysis uses implicit Euler in time and finite differences in space.Source: §1 · p. 3

2. The Model problem

The model problem is a pair of coupled one-dimensional linear heat equations on adjoining domains with different material parameters. The paper formulates the interface conditions and the DNWR discretization used for its analysis.

2. The Model problem

  • The model consists of heat equations on two one-dimensional domains, with material parameters, forcing, initial data, and boundary data specified for each subproblem.The domains are Q1 = (−a, 0) and Q2 = (0, b).Source: §2 · p. 4
  • At the interface, the model imposes continuity of temperature and heat flux between the two subdomains.The heat-flux condition weights each temperature gradient by its domain’s conductivity.Source: §2 · p. 4
  • The fully discrete setup uses N implicit Euler steps and centered finite differences on separate spatial grids, then applies DNWR with a relaxation step.The Neumann condition is discretized using a ghost point.Source: §2 · pp. 4–6

3. Error estimates for fully discrete DNWR

The section derives finite-time error estimates for fully discrete DNWR using an exponentially weighted time-discrete Fourier transform. It also presents a linear estimate and discusses how the time-window length affects the estimate.

3. Error estimates for fully discrete DNWR

  • The analysis applies an exponentially weighted Fourier transform and extends the iteration sequences in time so the transform can be used.For the stated lemma, the sequence is square-summable and zero for negative time indices.Source: §3 · pp. 6–7
  • The transformed subproblem difference equations are solved analytically to obtain expressions used in the error estimate.The derivation imposes the subproblem boundary conditions and uses induction on the iteration.Source: §3 · pp. 7–8
  • Theorem 1 gives an error estimate for the fully discrete DNWR iteration on the finite interval [0, T].The proof uses the exponentially weighted discrete Fourier transform and Parseval’s theorem.Source: §3 · p. 8
  • Setting the weight parameter r to zero yields a linear error estimate for the fully discrete iteration.This is stated as Corollary 1.Source: §3 · p. 8
  • For large T, the exponential factor exp(rT) dominates and the optimal r is close to zero, yielding linear convergence.For small T, exp(rT) is approximately 1, and the optimal r is close to the minimizer of the remaining estimate expression.Source: §3 · p. 9

4. Analysis of the fully discrete convergence estimate

Theorem 2 separates fully discrete convergence into CFL-dependent regimes: large subsolver CFL numbers yield behavior close to the time-discrete estimate, while small CFL numbers make spatial discretization important. The case studies compare fully discrete and time-discrete convergence factors across frequency and weight choices.

4. Analysis of the fully discrete convergence estimate

  • Theorem 2 analyzes the regimes |s_m| ≪1 and |s_m| ≫1, with |s_m| bounded by the inverse CFL number in each subsolver.The inverse-CFL bound links the theorem’s asymptotic cases to the solvers’ CFL numbers.Source: §4 · p. 9, §4.1 · p. 10

4.1. Discussion of Theorem 2.

  • For c_m ≫1, spatial discretization does not affect convergence, and the fully discrete estimate is close to the time-discrete estimate.The estimate is therefore independent of Δx1 and Δx2 in this regime.Source: §4.1 · p. 10
  • For c_m ≪ 1, fully discrete estimates are expected to converge linearly; with Δx_m = 0.1 and m = 1, 2, the high-frequency fully discrete convergence factor approaches 2/3.Source: §4.1 · p. 10, §4.2 · p. 12

4.2. Case studies.

  • In the frequency study, fully discrete and time-discrete factors are almost identical at large CFL numbers, but decreasing CFL causes them to deviate.The reported deviation begins at high frequencies as the values of s_m decrease.Source: §4.2 · pp. 11–12, Figure 1 · p. 11
  • For ω = 0, the convergence factor can cross zero as r varies because the factor changes sign.The figure compares fully discrete and time-discrete factors as functions of r.Source: §4.2 · p. 12, Figure 2 · p. 12

5. Numerical experiments

The experiments show that fully discrete estimates track the observed DNWR errors across grid and time-step choices. The behavior varies with the subsolvers’ CFL numbers: large CFL numbers yield grid-insensitive results, while small CFL numbers show mesh-dependent linear convergence.

Numerical procedure

  • The numerical comparison uses randomly initialized waveform error and computes the estimate’s min-max problem with SciPy’s Nelder–Mead routine at 10^-6 absolute tolerance.For linearly convergent cases, the reported rate averages the error ratio over the first five iterations.Source: §5 · p. 12

Simple

  • With large subsolver CFL numbers, the Simple-case fully discrete estimate tracks the normalized error and remains almost unchanged across the tested grid sizes.The fully discrete and time-discrete estimates are also nearly identical for ∆x = 0.01 and 0.0025.Source: §5 · p. 13, Figure 3 · p. 13
  • The Simple-case time-discrete and fully discrete estimates are superlinear for the first few iterations before becoming linear, whereas the continuous estimate remains superlinear.The continuous estimate therefore separates from the other two after a few iterations.Source: §5 · p. 14
  • At ∆t = 10^-6, the Simple-case time-discrete estimate nearly matches the continuous estimate, while the fully discrete estimate accurately predicts linear error convergence.The subsolver CFL numbers are 6.4 and 0.1, placing this case in a transition region; smaller ∆x brings the fully discrete curves closer to the time-discrete estimate.Source: §5 · p. 14, Figure 4 · p. 14
  • For ∆x = 0.0025, estimated and observed rates are 0.48 and 0.54; for ∆x = 0.01, they are 0.60 and 0.65, close to the small-CFL rate 2/3.The latter rates suggest small-CFL behavior for ∆x = 0.01.Source: §5 · p. 14

Air-Steel

  • For Air-Steel, the fully discrete estimate captures the slower normalized-error convergence, while continuous and time-discrete estimates reach below 10^-9 in one iteration.For the tested meshes, estimated and observed convergence rates are close; the largest difference is a factor of 2 at ∆x1 = ∆x2 = 10^-4.Source: Figure 5 · p. 15, §5 · p. 15, Table 1 · p. 15

6. Conclusion

The paper derives fully discrete l2 error estimates for DNWR applied to coupled heat equations. The estimates distinguish large- and small-CFL convergence behavior, and numerical experiments validate their accuracy across varied grids, time steps, and time horizons.

Conclusion

  • Exponentially weighted Fourier analysis with an auxiliary weight r yields l2 error estimates depending on T, ∆t, ∆x1, and ∆x2 for implicit Euler and finite-difference discretizations.The estimates concern DNWR for two coupled heat equations.Source: §6 · p. 15
  • For large subsolver CFL numbers, convergence matches the time-discrete counterpart and is independent of ∆x1 and ∆x2; for small CFL numbers, linear convergence depends on ∆x1/∆x2.Source: §6 · p. 16
  • Numerical experiments show that fully discrete estimates accurately capture observed DNWR convergence across a wide range of T, ∆t, ∆x1, and ∆x2.Source: §6 · p. 16
Loading 2610.01475v1…