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Higher order Galerkin methods in time for free surface flows
Author(s) -
Weller Stephan
Publication year - 2014
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.201410408
Subject(s) - discretization , galerkin method , mathematics , free surface , decoupling (probability) , mathematical analysis , flow (mathematics) , surface tension , convergence (economics) , capillary action , stability (learning theory) , geometry , physics , finite element method , mechanics , computer science , control engineering , machine learning , engineering , economics , thermodynamics , economic growth , quantum mechanics
In many engineering applications, free surface or two‐phase flows are discretized in time with an explicit decoupling of geometry and fluid flow. Such a strategy leads to a capillary CFL condition of the form $\Delta t \le \sqrt{\rm{We}}\,h^{3/2}$ [3]. For the case of surface tension dominated flows (i.e. high Weber number We) this can dictate infeasibly small time steps. As an alternative we suggest a Galerkin method in time based on the discontinuous Galerkin method of first order (dG(1)). For this choice, an energy estimate can be proved [7], so unconditional stability of the method is given. While for ODEs or parabolic PDEs the method is of third order at the discrete points in time t n [4], in the case of free surface flows second order convergence can still be achieved. Numerical examples using the Arbitrary Lagrangian Eulerian (ALE) method for both capillary one‐phase and two‐phase flow demonstrate this convergence order. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)