## Abstract

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(Nlog. N) 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.

Original language | English (US) |
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Pages (from-to) | 168-191 |

Number of pages | 24 |

Journal | Journal of Computational Physics |

Volume | 247 |

DOIs | |

State | Published - Aug 5 2013 |

## All Science Journal Classification (ASJC) codes

- Numerical Analysis
- Modeling and Simulation
- Physics and Astronomy (miscellaneous)
- General Physics and Astronomy
- Computer Science Applications
- Computational Mathematics
- Applied Mathematics

## Keywords

- Boundary integral method
- Ewald summation
- Interfacial flow
- Surface tension