A Splitting Method for Numerical Simulation of Free Surface Flows of Incompressible Fluids with Surface Tension

Kirill D. Nikitin 1 , Maxim A. Olshanskii 2 , Kirill M. Terekhov 3 ,  and Yuri V. Vassilevski 4
  • 1 Institute of Numerical Mathematics, Russian Academy of Sciences, Gubkin str. 8, 119333 Moscow, Russia
  • 2 Department of Mathematics, University of Houston, 641 PGH Building, Houston, TX 77204-3008, USA
  • 3 Department of Energy Resources Engineering, Stanford University, 367 Panama Street, Stanford, CA 94305-2220, USA; and Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, Russia
  • 4 Institute of Numerical Mathematics, Russian Academy of Sciences, Gubkin str. 8, 119333 Moscow; and Moscow Institute of Physics and Technology, Institutskii Lane 9, 141700, Dolgoprudny, Russia


The paper studies a splitting method for the numerical time-integration of the system of partial differential equations describing the motion of viscous incompressible fluid with free boundary subject to surface tension forces. The method splits one time step into a semi-Lagrangian treatment of the surface advection and fluid inertia, an implicit update of viscous terms and the projection of velocity into the subspace of divergence-free functions. We derive several conservation properties of the method and a suitable energy estimate for numerical solutions. Under certain assumptions on the smoothness of the free surface and its evolution, this leads to a stability result for the numerical method. Efficient computations of free surface flows of incompressible viscous fluids need several other ingredients, such as dynamically adapted meshes, surface reconstruction and level set function re-initialization. These enabling techniques are discussed in the paper as well. The properties of the method are illustrated with a few numerical examples. These examples include analytical tests and the oscillating droplet benchmark problem.

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CMAM considers original mathematical contributions to computational methods and numerical analysis with applications mainly related to PDEs. The journal is interdisciplinary while retaining the common thread of numerical analysis, readily readable and meant for a wide circle of researchers in applied mathematics.