arXiv:2611.05428v1 Announce Type: cross Abstract: The method of regularized Stokeslets (MRS) is widely used to model fluid-structure interactions of microscopic bodies in viscous environments. Specifically, MRS computes the fluid velocity generated by regularized forces distributed on an immersed body, which in turn defines a system of ordinary differential equations with which to evolve the body so that the no-slip condition is enforced. To accurately model material properties such as elasticity and bending rigidity, the resulting ODE system is often stiff and, therefore, computationally expensive to solve with explicit time-integration methods. Here we demonstrate the potential of linearly implicit Rosenbrock (ROS) time integration methods to greatly reduce the computational time required to solve such ODE systems. We focus on the classic model problem of the swimming of a flexible filament driven by preferred curvature forcing. Our findings indicate that, compared to previous MRS simulations, the increased stability of ROS methods allows for speed-ups between twenty and two thousand times over explicit Runge-Kutta (ERK) methods. Moreover, we present approximate scaling laws for this speedup as a function of the key model parameters (bending rigidity and slenderness) at the root of numerical stiffness. Lastly, beyond the basic ROS approach, we discuss several additional strategies that further improve performance.