A small-scale decomposition for 3D boundary integral computations with surface tension

An efficient, non-stiff boundary integral method for the initial value problem for interfacial Darcy flow (which is a model of porous media flow) in three space dimensions is presented. We consider a ‘doubly-periodic’ interface separating two fluids, with surface tension present at the boundary. Surface tension introduces high order (i.e., high derivative) terms in the governing equation, and this imposes a severe stability constraint on explicit time-integration methods. Furthermore, the high order terms appear in a nonlocal operator, which makes it difficult to design an efficient implicit method. The stiffness is removed by developing a small-scale decomposition in the spirit of prior work in the two-dimensional problem by Hou, Lowengrub, and Shelley. In order to develop this small-scale decomposition, we formulate the problem using a generalized isothermal parameterization of the free surface. An additional difficulty is the efficient calculation of the Birkhoff-Rott integral for the velocity of the interface. We present a new algorithm, based on Ewald summation, to compute this in O(N logN) operations, where N is the number of interface grid points. Our non-stiff method is expected to apply widely to problems for doubly-periodic interfacial flow with surface tension which have a boundary integral formulation.