CONVERGENCE OF A NON-STIFF BOUNDARY INTEGRAL METHOD FOR INTERFACIAL FLOWS WITH SURFACE-TENSION

Boundary integral methods to simulate interfacial flows are very sensi tive to numerical instabilities. In addition, surface tension introduc es nonlinear terms with high order spatial derivatives into the interf ace dynamics. This makes the spatial discretization even more difficul t and, at the same time, imposes a severe time step constraint for sta ble explicit time integration methods. A proof of the convergence of a reformulated boundary integral method for two-density fluid interface s with surface tension is presented. The method is based on a scheme i ntroduced by Hou, Lowengrub and Shelley [J. Comp. Phys. 114 (1994), pp . 312-338] to remove the high order stability constraint or stiffness. Some numerical filtering is applied carefully at certain places in th e discretization to guarantee stability. The key of the proof is to id entify the most singular terms of the method and to show, through ener gy estimates, that these terms balance one another. The analysis is at a time continuous-space discrete level but a fully discrete case for a simple Hele-Shaw interface is also studied. The time discrete analys is shows that the high order stiffness is removed and also provides an estimate of how the CFL constraint depends on the curvature and regul arity of the solution. The robustness of the method is illustrated wit h several numerical examples. A numerical simulation of an unstably st ratified two-density interfacial flow shows the roll-up of the interfa ce; the computations proceed up to a time where the interface is about to pinch off and trapped bubbles of fluid are formed. The method rema ins stable even in the full nonlinear regime of motion. Another applic ation of the method shows the process of drop formation in a falling s ingle fluid.