Analysis of Schwarz waveform relaxation for the coupled Ekman boundary layer problem with continuously variable coefficients

Thery, Sophie; Pelletier, Charles; Lemarié, Florian; Blayo, Éric

doi:10.1007/s11075-021-01149-y

Cited by 5 publications

(6 citation statements)

References 29 publications

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“…We examine the solutions 𝑈 𝑎 , 𝑈 𝑜 of coupled 1D linear reaction-diffusion equations which is a proxy for coupled ocean-atmosphere problems [3,5]:…”

Section: Simplified Air-sea Coupled Problemmentioning

confidence: 99%

Semi-Discrete Analysis of a Simplified Air-Sea Coupling Problem with Nonlinear Coupling Conditions

Clement,

Lemarie,

Blayo

2024

Lecture Notes in Computational Science and Engineering

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“…We examine the solutions 𝑈 𝑎 , 𝑈 𝑜 of coupled 1D linear reaction-diffusion equations which is a proxy for coupled ocean-atmosphere problems [3,5]:…”

Section: Simplified Air-sea Coupled Problemmentioning

confidence: 99%

Semi-Discrete Analysis of a Simplified Air-Sea Coupling Problem with Nonlinear Coupling Conditions

Clement,

Lemarie,

Blayo

2024

Lecture Notes in Computational Science and Engineering

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“…In this subsection, the objective is to incorporate in the convergence analysis the impact of the time discretisation. The error in time will now be interpreted as a discrete signal {e(n)} ∞ n=0 with constant 1 For finite domains of size H, [27] gives ρ…”

Section: Time Discretisationmentioning

confidence: 99%

Discrete analysis of Schwarz waveform relaxation for a diffusion reaction problem with discontinuous coefficients

Clément¹,

Lemarié²,

Blayo³

2022

The SMAI Journal of Computational Mathematics

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In this paper, we investigate the effect of the space and time discretisation on the convergence properties of Schwarz Waveform Relaxation (SWR) algorithms. We consider a reaction-diffusion problem with discontinuous coefficients discretised on two non-overlapping domains with several numerical schemes (in space and time). A methodology to determine the rate of convergence of the classical SWR method with standard interface conditions (Dirichlet-Neumann or Robin-Robin) accounting for discretisation errors is presented. We discuss how such convergence rates differ from the ones derived at a continuous level (i.e. assuming an exact discrete representation of the continuous problem). In this work we consider a second-order finite difference scheme and a finite volume scheme based on quadratic spline reconstruction in space, combined with either a simple backward Euler scheme or a twostep "Padé" scheme (resembling a Diagonally Implicit Runge Kutta scheme) in time. We prove those combinations of space-time schemes to be unconditionally stable on bounded domains. We illustrate the relevance of our analysis with specifically designed numerical experiments.

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“…where 0 is any first guess extended to infinite time. Using the particular extension 0 | [0, ] = e 0 and 0 | ] ,∞[ = 0 leads to (10). Then combining e ( , •) 2 / e 0 ( , •) 2 = Π =1 , ( ) and ( 10) leads to (11).…”

Section: Difficulties Expressing Error Over a Finite Time Windowmentioning

confidence: 99%

“…Remark 2 A bound on the convergence factor given by ( 9) was already calculated in [10]. This bound is complicated to calculate and then hardly usable.…”

Section: Difficulties Expressing Error Over a Finite Time Windowmentioning

confidence: 99%

On the Links Between Observed and Theoretical Convergence Rates for Schwarz Waveform Relaxation Algorithm for the Time-Dependent Problems

Thery

2022

Lecture Notes in Computational Science and Engineering

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In the framework of linear problems, a usual approach to study the convergence of Schwarz algorithms is to calculate the convergence rate in the frequency domain. However, for time-dependent problems, this tool provides results not fully representative of the observed behaviour of the algorithm. In this article we highlight differences between the theoretical convergence rate and the convergence observed in the physical space-time domain. We also explain how the theoretical convergence rate can be used to provide bounds to the observed convergence rate obs . For problems defined on time windows of finite size, we recall that the bounds usually considered to study the convergence are empirical estimates albeit robust. In conclusion of this paper, numerical experiments are carried out to illustrate the relevance of the theoretical analysis.

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Analysis of Schwarz waveform relaxation for the coupled Ekman boundary layer problem with continuously variable coefficients

Cited by 5 publications

References 29 publications

Semi-Discrete Analysis of a Simplified Air-Sea Coupling Problem with Nonlinear Coupling Conditions

Semi-Discrete Analysis of a Simplified Air-Sea Coupling Problem with Nonlinear Coupling Conditions

Discrete analysis of Schwarz waveform relaxation for a diffusion reaction problem with discontinuous coefficients

On the Links Between Observed and Theoretical Convergence Rates for Schwarz Waveform Relaxation Algorithm for the Time-Dependent Problems

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