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Convergence of a non-stiff boundary integral method for interfacial flows with surface tension

Authors: Héctor D. Ceniceros and Thomas Y. Hou
Journal: Math. Comp. 67 (1998), 137-182
MSC (1991): Primary 65M12, 76B15
MathSciNet review: 1443116
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Abstract: Boundary integral methods to simulate interfacial flows are very sensitive to numerical instabilities. In addition, surface tension introduces nonlinear terms with high order spatial derivatives into the interface dynamics. This makes the spatial discretization even more difficult and, at the same time, imposes a severe time step constraint for stable explicit time integration methods. A proof of the convergence of a reformulated boundary integral method for two-density fluid interfaces with surface tension is presented. The method is based on a scheme introduced 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 the discretization to guarantee stability. The key of the proof is to identify the most singular terms of the method and to show, through energy 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 analysis shows that the high order stiffness is removed and also provides an estimate of how the CFL constraint depends on the curvature and regularity of the solution. The robustness of the method is illustrated with several numerical examples. A numerical simulation of an unstably stratified two-density interfacial flow shows the roll-up of the interface; the computations proceed up to a time where the interface is about to pinch off and trapped bubbles of fluid are formed. The method remains stable even in the full nonlinear regime of motion. Another application of the method shows the process of drop formation in a falling single fluid.

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Additional Information

Héctor D. Ceniceros
Affiliation: Department of Applied Mathematics, California Institute of Technology, Pasadena, California 91125
Address at time of publication: Centro de Investigación en Computación Instituto Politécnico Nacional Col. Lindavista, Mexico City, Mexico 07300.

Thomas Y. Hou
Affiliation: Department of Applied Mathematics, California Institute of Technology, Pasadena, California 91125

Keywords: boundary integral method, surface tension, stiffness.
Received by editor(s): December 7, 1995
Received by editor(s) in revised form: June 5, 1996
Additional Notes: The first author was partially supported by the Office of Naval Research under Grant N00014-94-1-0310.
The second author was partially supported by the Office of Naval Research under Grant N00014-94-1-0310 and the National Science Foundation under grant DMS-9407030.
Article copyright: © Copyright 1998 American Mathematical Society