Fully implicit time discretization for a free surface flow problem

Abstract

AbstractOne of the challenges in the numerics of free surface flows is the coupling of the flow field to the geometry of the domain. The most simple approach is an explicit decoupling, i.e. computing the flow field with geometrical information of a prior time step and then updating the geometry. This widely used approach leads to a severe CFL condition of the type $\Delta t \le C\sqrt{{\rm We}}\;h^{3 \over 2}$, which may prescribe infinitesimally small time step sizes in the interesting case of a small Weber number (i.e. high surface tension). A semi‐implicit approach utilizing the fact that $x^{n+1} = x^n + \Delta t\, u^{n+1}_{|\Gamma}$, where xk is a parametrization of the capillary boundary Γ, is also available [1]. This approach can be proven to be unconditionally stable but is of first order only. It also suffers from relatively strong numerical dissipation.We present a fully implicit approach using a backward differentiation formula to achieve a time discretization method that is of second order and only minimally dissipative. A numerical example of an oscillating drop showing very low numerical dissipation and second order convergence as well as numerical evidence for the stability of the method is presented. Since the method requires the solution of a highly nonlinear coupled system, possible preconditioners for this system are discussed, including a lower order decoupling. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)