We propose finite-volume schemes for the Cahn-Hilliard equation which unconditionally and discretely preserve the boundedness of the phase field and the dissipation of the free energy.Our numerical framework is applic...We propose finite-volume schemes for the Cahn-Hilliard equation which unconditionally and discretely preserve the boundedness of the phase field and the dissipation of the free energy.Our numerical framework is applicable to a variety of free-energy potentials,including Ginzburg-Landau and Flory-Huggins,to general wetting boundary conditions,and to degenerate mobilities.Its central thrust is the upwind methodology,which we combine with a semi-implicit formulation for the freeenergy terms based on the classical convex-splitting approach.The extension of the schemes to an arbitrary number of dimensions is straightforward thanks to their dimensionally split nature,which allows to efficiently solve higher-dimensional problems with a simple parallelisation.The numerical schemes are validated and tested through a variety of examples,in different dimensions,and with various contact angles between droplets and substrates.展开更多
基金supported by Labex CEMPI(ANR-11-LABX-0007-01).RBJAC were supported by the ERC Advanced Grant No.883363(Nonlocal PDEs for Complex Particle Dynamics(Nonlocal-CPD):Phase Transitions,Patterns and Synchronization)under the European Union’s Horizon 2020 research and innovation programme+2 种基金JAC was partially supported by EPSRC Grants No.EP/V051121/1(Stability analysis for non-linear partial differential equations across multiscale applications)under the EPSRC lead agency agreement with the NSF,and EP/T022132/1(Spectral element methods for fractional differential equations,with applications in applied analysis and medical imaging)SK was partially supported by EPSRC Platform Grant No.EP/L020564/1(Multiscale Analysis of Complex Interfacial Phenomena(MACIPh):Coarse graining,Molecular modelling,stochasticity,and experimentation)EPSRC Grant No.EP/L027186/1(Fluid processes in smart microengineered devices:Hydrodynamics and thermodynamics in microspace).SPP acknowledges financial support from the Imperial College President’s PhD Scholarship scheme.
文摘We propose finite-volume schemes for the Cahn-Hilliard equation which unconditionally and discretely preserve the boundedness of the phase field and the dissipation of the free energy.Our numerical framework is applicable to a variety of free-energy potentials,including Ginzburg-Landau and Flory-Huggins,to general wetting boundary conditions,and to degenerate mobilities.Its central thrust is the upwind methodology,which we combine with a semi-implicit formulation for the freeenergy terms based on the classical convex-splitting approach.The extension of the schemes to an arbitrary number of dimensions is straightforward thanks to their dimensionally split nature,which allows to efficiently solve higher-dimensional problems with a simple parallelisation.The numerical schemes are validated and tested through a variety of examples,in different dimensions,and with various contact angles between droplets and substrates.
