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On the stability of a semi-implicit scheme of Cahn-Hilliard type equations Cheng, Xinyu


It is well known that Allen-Cahn equation and Cahn-Hilliard equation are essential to study the phase separation phenomenon of a two-phase or a multiple-phase mixture. An important property of the solutions to those two equations is that the energy functional, which is defined in this thesis, decreases in time. To study these solutions, researchers developed different numerical schemes to give accurate approximations, since analytic solutions are only available in a very few simple cases. However, not all schemes satisfy the energy-decay property, which is an important standard to determine whether the scheme is stable. In recent work, Li, Qiao and Tang developed a semi-implicit scheme for the Cahn-Hilliard equation and proved the energy-decay property. In this thesis, we extend the semi-implicit scheme to the Allen-Cahn equation and fractional Cahn-Hilliard equation with a proof of the energy-decay property. Moreover, this semi-implicit scheme is practical and could be applied to more general diffusion equations while preserving the energy-decay stability.

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