Communications in Mathematical Sciences

Volume 2 (2004)

Number 1

Conservative multigrid methods for ternary Cahn-Hilliard systems

Pages: 53 – 77



Kyungkeun Kang

Junseok Kim

John Lowengrub


We develop a conservative, second order accurate fully implicit discretization of ternary (three-phase) Cahn-Hilliard (CH) systems that has an associated discrete energy functional. This is an extension of our work for two-phase systems. We analyze and prove convergence of the scheme. To efficiently solve the discrete system at the implicit time-level, we use a nonlinear multigrid method. The resulting scheme is efficient, robust and there is at most a 1st order time step constraint for stability. We demonstrate convergence of our scheme numerically and we present several simulations of phase transitions in ternary systems.


Ternary Cahn-Hilliard system; nonlinear multigrid method

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