Error analysis of a stochastic immersed boundary method incorporating thermal fluctuations

A stochastic numerical scheme for an extended immersed boundary method which incorporates thermal fluctuations for the simulation of microscopic biological systems consisting of fluid and immersed elastica was introduced in reference [2]. The numerical scheme uses techniques from stochastic calculus to overcome stability and accuracy issues associated with standard finite difference methods. The numerical scheme handles a range of time steps in a unified manner, including time steps which are greater than the smallest time scales of the system. The time step regimes we shall investigate can be classified as small, intermediate, or large relative to the time scales of the fluid dynamics of the system. Small time steps resolve in a computationally explicit manner the dynamics of all the degrees of freedom of the system. Large time steps resolve in a computationally explicit manner only the degrees of freedom of the immersed elastica, with the contributions of the dynamics of the fluid degrees of freedom accounted for in only a statistical manner over a time step. Intermediate time steps resolve in a computationally explicit manner only some degrees of freedom of the fluid with the remaining degrees of freedom accounted for statistically over a time step. In this paper, uniform bounds are established for the strong error of the stochastic numerical method for each of the time step regimes. The scaling of the numerical errors with respect to the parameters of the method is then discussed.