Numerical stability analysis for an immersed elastic surface using the method of regularized Stokeslets

The method of regularized Stokeslets (MRS) is a widely used numerical method for simulating immersed elastic structures in viscous fluids at zero Reynolds number. While straightforward to implement, a numerical challenge arises in the form of stiffness, especially for elastic stuctures with both tensile and bending rigidity. Here, we present the first linear stability analysis of a fluid-structure interaction problem using the MRS. Applying small perturbations to a doubly-periodic elastic surface immersed in a 3D fluid, we solve the resultant linearized system with Fourier analysis. The eigenvalues of the system determine the theoretical critical time step for numerical stability for a forward Euler time integration. This is verified numerically for several regularization functions, elastic models, and parameter choices. New doubly periodic regularized Stokeslets are presented, allowing for comparison of the stability properties of different regularization functions. The stability results for a common regularization function are approximated by a power law relating the regularization parameter and the surface discretization for two different elastic models. This relationship is empirically shown to hold in the different setting of a finite surface in a bulk fluid.