Energy-based operator splitting approach for the time discretization of coupled systems of partial and ordinary differential equations for fluid flows: The Stokes case

TL;DR

This paper introduces an energy-preserving operator splitting method for coupled Stokes and ODE systems, ensuring unconditional stability in fluid flow simulations without iterative coupling.

Contribution

The paper presents a novel energy-based splitting scheme that maintains physical energy balance and guarantees unconditional stability for coupled Stokes-ODE problems.

Findings

The method is unconditionally stable regardless of time step size.

Stability and convergence are validated on three analytical test cases.

The approach avoids iterative coupling between Stokes and ODE substeps.

Abstract

The goal of this work is to develop a novel splitting approach for the numerical solution of multiscale problems involving the coupling between Stokes equations and ODE systems, as often encountered in blood flow modeling applications. The proposed algorithm is based on a semi-discretization in time based on operator splitting, whose design is guided by the rationale of ensuring that the physical energy balance is maintained at the discrete level. As a result, unconditional stability with respect to the time step choice is ensured by the implicit treatment of interface conditions within the Stokes substeps, whereas the coupling between Stokes and ODE substeps is enforced via appropriate initial conditions for each substep. Notably, unconditional stability is attained without the need of subiterating between Stokes and ODE substeps. Stability and convergence properties of the proposed algorithm are tested on three specific examples for which analytical solutions are derived.