Spatially Adaptive Stochastic Multigrid Methods for Fluctuating Hydrodynamics

The immersed boundary method is a numerical approach for simulating elastic structures which interact with a fluid flow. In many physical systems thermal fluctuations become significant at small scales and play a fundamental role. In this paper stochastic numerical methods are developed which extend the immersed boundary approach to account for thermal fluctuations by including appropriate stochastic forcing terms in the fluid equations. The stochastic numerical methods developed in this paper differ in three significant ways from prior work: (i) The new numerical methods allow for use of non-periodic non-uniform multilevel meshes, where prior methods were only applicable to uniform periodic meshes and relied heavily on the Fourier Transform. (ii) A new stochastic closure approximation is derived for the fast dynamics of the system to handle stiff features of the stochastic equations. (iii) Methods for the generation of stochastic fields with long-range covariance structure on multilevel meshes are developed having only linear computational complexity in the number of mesh cells. These advances in addition to allowing for improved accuracy and computational efficiency also allow for new physical phenomena to be studied with the stochastic immersed boundary method. To show how the methods can be used in practice, results for an interacting particle system and a polymer system are discussed which make particular use of non-periodic boundary conditions to capture in the hydrodynamic interactions the effects of walls and fixed inclusions in the fluid.