Rigorous embedding of cell dynamics simulations in the Cahn-Hilliard-Cook framework: Imposing stability and isotropy

Area of Science:

• Computational physics and materials science.
• Numerical methods for partial differential equations.

Background:

• The cell dynamics simulations (CDS) method is efficient but suffers from grid artifacts.
• Theoretical understanding of CDS stability has been limited for decades.
• The local averaging operator in CDS lacks optimal isotropy and scaling properties.

Purpose of the Study:

• To rigorously analyze the stability of the cell dynamics simulations (CDS) method.
• To provide a practical method for improving CDS by relating it to finite-difference approximations.
• To suppress unphysical grid artifacts in CDS results.

Main Methods:

• Analysis of the local averaging operator in matrix form.
• Relating CDS to finite-difference approximations of Cahn-Hilliard-Cook equations.
• Quantitative comparison of isotropy and scaling behavior of discrete Laplacians.

Main Results:

• A direct link between CDS and Cahn-Hilliard-Cook equations was established.
• Replacing CDS averaging operators with discrete Laplacians suppresses grid artifacts.
• Discrete Laplacians from Oono-Puri and BvV methods offer superior isotropy and scaling.

Conclusions:

• The stability of CDS can be enhanced by using specific discrete Laplacians.
• Proper rescaling of discrete Laplacians is equivalent to choosing an optimal time step in CDS.
• This work provides stability conditions for phase-field simulations and identifies optimal discrete Laplacians for field-based methods.