[Paper Review] Stability and error analysis of a splitting method using Robin-Robin coupling applied to a fluid-structure interaction problem

Erik Burman, R. Durst+1 more|arXiv (Cornell University)|Nov 15, 2019

Advanced Numerical Methods in Computational Mathematics19 references5 citations

TL;DR

This paper presents a stability and error analysis of a Robin-Robin splitting method for fluid-structure interaction (FSI) problems, using an explicit, loosely coupled scheme with Robin transmission conditions at the fluid-structure interface. The key contribution is a rigorous error estimate showing the global error at time $T$ is $O(\sqrt{T\Delta t})$, independent of spatial discretization, when the Robin parameter $\lambda$ is $O(1)$, proving convergence without requiring $h^{-1}$-scaled stabilization terms.

ABSTRACT

We analyze a splitting method for a canonical fluid structure interaction problem. The splittling method uses a Robin-Robin boundary condition, explicit strategy. We prove the method is stable and, furthermore, we provide an error estimate that shows the error at the final time $T$ is $O(\sqrt{TΔt})$ where $Δt$ is the time step.

Motivation & Objective

• To address the instability of classical loosely coupled schemes in fluid-structure interaction due to the added mass effect.
• To analyze a Robin-Robin coupling method without additional pressure stabilization, which had been observed numerically to be stable but lacked theoretical foundation.
• To establish a rigorous error estimate for the time-discrete splitting scheme, showing convergence independent of spatial discretization.
• To remove the need for $h^{-1}$-scaled stabilization terms that previously limited time step size and required iterative corrections.

Proposed method

• The method employs a time-splitting strategy where the fluid and solid PDEs are solved sequentially on each time step using Robin-Robin transmission conditions at the interface.
• The Robin-Robin coupling uses interface fluxes involving normal velocity and traction, with a parameter $\lambda$ that controls the coupling strength.
• The analysis assumes sufficient regularity of the local PDE solutions and avoids spatial discretization, focusing on time discretization error.
• A key technique involves energy estimates using a modified energy norm and a method inspired by G. Baker to avoid exponential growth in error bounds.
• The truncation error is derived by comparing the exact solution to the splitting scheme, and combined with stability to bound the global error.
• A parameter $\delta = \Delta t / (2T)$ is used in Young's inequality to control error accumulation over time steps.

Experimental results

Research questions

• RQ1 Is the Robin-Robin loosely coupled scheme stable without additional pressure stabilization for fluid-structure interaction?
• RQ2 Can a rigorous error estimate be derived for the time-discrete Robin-Robin splitting method, independent of spatial discretization?
• RQ3 Does the error grow exponentially or like $\sqrt{T\Delta t}$ over time, and can this be proven using energy methods?
• RQ4 Can the $O(\sqrt{T\Delta t})$ error bound be achieved with $\lambda = O(1)$, avoiding the need for $h^{-1}$-scaling of stabilization parameters?
• RQ5 Is convergence of the time-discrete scheme guaranteed under this formulation, even without iterative correction?

Key findings

• The global error of the Robin-Robin splitting scheme is bounded by $O(\sqrt{T\Delta t})$ in a suitable energy norm, with the constant independent of spatial discretization.
• The error estimate grows as $\sqrt{T}$ rather than exponentially, which is a significant improvement over previous methods.
• The analysis proves stability and convergence of the time-discrete scheme without requiring any pressure stabilization or $h^{-1}$-scaling of the Robin parameter.
• The method achieves convergence with $\lambda = O(1)$, eliminating the need for very small time steps or iterative correction procedures.
• The error bound is derived using a novel energy technique that avoids exponential growth, inspired by G. Baker’s approach.
• The result implies that the method is robust for long-time simulations and allows for efficient spatial discretization choices.