Rosenbrock Integrators Deliver 20-2000x Speedups in Regularized Stokeslet Filament Models

The preprint applies Rosenbrock-Wanner schemes to the method of regularized Stokeslets for a preferred-curvature-driven filament, converting the no-slip-enforced velocity equations into a linearly implicit system that tolerates far larger steps than explicit integrators when bending rigidity produces stiffness. Tests across slenderness ratios from 0.01 to 0.001 and rigidity values spanning three orders of magnitude show the speedup grows roughly linearly with rigidity and inversely with slenderness, confirming the method removes the dominant stability bottleneck without loss of spatial accuracy.

Prior explicit Runge-Kutta implementations in the same MRS framework required step-size reductions that scaled poorly with the fourth-power dependence of bending resistance on filament radius; the new approach bypasses that restriction while preserving the O(N) cost per step typical of fast multipole or Ewald summation Stokeslet evaluations. This directly benefits microswimmer and cilia simulations used in synthetic biology and microfluidic device design, where ensemble runs over hundreds of parameter combinations are routine.

The work stops short of full three-dimensional body geometries or distributed actuation, leaving open whether the same gains hold once nonlocal hydrodynamic interactions dominate. Follow-on studies that embed Rosenbrock time-stepping inside adaptive-mesh or boundary-integral codes for eukaryotic flagella would test whether the reported scaling persists at higher Reynolds-number transitions or with nonlinear constitutive laws.