<p>The numerical simulation of incompressible fluid-structure interaction systems with loosely coupled schemes is a delicate problem. Indeed, the splitting method must both be stable for the full nonlinear system and have sufficient accuracy to be of use in practice. In the case of the coupling of an incompressible fluid with thick-walled solids the error analyses reported in the literature are limited to <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_2939_Article_IEq1.gif" Format="GIF" Height="24" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {O}}(\tau ^{\frac{1}{2}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <msup> <mi>τ</mi> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> accuracy in time, where <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_2939_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="11" /> </InlineMediaObject> <EquationSource Format="TEX">$\tau $</EquationSource> <EquationSource Format="MATHML"><math> <mi>τ</mi> </math></EquationSource> </InlineEquation> denotes the time step. The objective of the present work is to show two important extensions of the analysis of the Robin-Robin loosely coupled scheme recently reported in <i>[Numer. Math., 151(4):807-840, 2022]</i>. First, we give a formulation of the scheme in a general non-linear setting and prove its unconditional energy stability. Then we show that nearly-optimal <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_2939_Article_IEq3.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="134" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {O}}\big (\tau \sqrt{1+\log \tau ^{-1}}\big )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <mi>τ</mi> <msqrt> <mrow> <mn>1</mn> <mo>+</mo> <mo>log</mo> <msup> <mi>τ</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> </msqrt> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> accuracy can be achieved in the linear case. These theoretical findings are illustrated in a series of numerical examples.</p>

Robin-Robin Loose Coupling for Incompressible Fluid-Structure Interaction: Non-Linear Setting and Nearly-Optimal Error Analysis

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发表日期：

2025年6月2日

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Erik Burman,
Rebecca Durst,
Miguel A. Fernández,
Johnny Guzmán,
Oscar Ruz

摘要

The numerical simulation of incompressible fluid-structure interaction systems with loosely coupled schemes is a delicate problem. Indeed, the splitting method must both be stable for the full nonlinear system and have sufficient accuracy to be of use in practice. In the case of the coupling of an incompressible fluid with thick-walled solids the error analyses reported in the literature are limited to ${\mathcal {O}}(\tau ^{\frac{1}{2}})$ $O (τ^{\frac{1}{2}})$ accuracy in time, where $\tau $ $τ$ denotes the time step. The objective of the present work is to show two important extensions of the analysis of the Robin-Robin loosely coupled scheme recently reported in [Numer. Math., 151(4):807-840, 2022]. First, we give a formulation of the scheme in a general non-linear setting and prove its unconditional energy stability. Then we show that nearly-optimal ${\mathcal {O}}\big (\tau \sqrt{1+\log \tau ^{-1}}\big )$ $O (τ \sqrt{1 + log τ^{- 1}})$ accuracy can be achieved in the linear case. These theoretical findings are illustrated in a series of numerical examples.