A non-stiff boundary integral method for 3D porous media flow with surface tension

We present an efficient, non-stiff boundary integral method for 3D porous media flow with surface tension. Surface tension introduces high order (i.e., high derivative) terms into the evolution equations, and this leads to severe stability constraints for explicit time-integration methods. Furthermore, the high order terms appear in non-local operators, making the application of implicit methods difficult. Our method uses the fundamental coefficients of the surface as dynamical variables, and employs a special isothermal parameterization of the interface which enables efficient application of implicit or linear propagator time-integration methods via a small-scale decomposition. The method is tested by computing the relaxation of an interface to a flat surface under the action of surface tension. These calculations employ an approximate interface velocity to test the stiffness reduction of the method. The approximate velocity has the same mathematical form as the exact velocity, but avoids the numerically intensive computation of the full Birkhoff–Rott integral. The algorithm is found to be effective at eliminating the severe time-step constraint that plagues explicit time-integration methods.