On an efficient numerical procedure for the Functionalized Cahn-Hilliard equation

Saulo Orizaga; Ogochukwu Ifeacho; Sampson Owusu; Saulo Orizaga; Ogochukwu Ifeacho; Sampson Owusu

doi:10.3934/math.20241010

AIMS Mathematics

2024, Volume 9, Issue 8: 20773-20792. doi: 10.3934/math.20241010

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On an efficient numerical procedure for the Functionalized Cahn-Hilliard equation

Department of Mathematics, New Mexico Tech, 801 Leroy Place, Socorro, NM 87801, USA

Received: 08 May 2024 Revised: 12 June 2024 Accepted: 20 June 2024 Published: 26 June 2024
MSC : 65T50, 65M99

The Functionalized Cahn Hilliard (FCH) equation was used to model micro-phase separation in mixtures of amphiphilic molecules in solvent. In this paper, we proposed a Tri-Harmonic Modified (THM) numerical approach for efficiently solving the FCH equation with symmetric double well potential by extending the ideas of the Bi-harmonic Modified (BHM) method. THM formulation allowed for the nonlinear terms in the FCH equation to be computed explicitly, leading to fast evaluations at every time step. We investigated the convergence properties of the new approach by using benchmark problems for phase-field models, and we directly compared the performance of the THM method with the recently developed scalar auxiliary variable (SAV) schemes for the FCH equation. The THM modified scheme was able to produce smaller errors than those obtained from the SAV formulation. In addition to this direct comparison with the SAV schemes, we tested the adaptability of our scheme by using an extrapolation technique which allows for errors to be reduced for longer simulation runs. We also investigated the adaptability of the THM method to other 6th order partial differential equations (PDEs) by considering a more complex form of the FCH equation with nonsymmetric double well potential. Finally, we also couple the THM scheme with a higher order time-stepping method, (implicit-explicit) IMEX schemes, to demonstrate the robustness and adaptability of the new scheme. Numerical experiments are presented to investigate the performance of the new approach.
- Cahn-Hilliard,
- functionalized Cahn-Hilliard,
- phase field Modeling,
- phase separation,
- BHM method
Citation: Saulo Orizaga, Ogochukwu Ifeacho, Sampson Owusu. On an efficient numerical procedure for the Functionalized Cahn-Hilliard equation[J]. AIMS Mathematics, 2024, 9(8): 20773-20792. doi: 10.3934/math.20241010

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Abstract

The Functionalized Cahn Hilliard (FCH) equation was used to model micro-phase separation in mixtures of amphiphilic molecules in solvent. In this paper, we proposed a Tri-Harmonic Modified (THM) numerical approach for efficiently solving the FCH equation with symmetric double well potential by extending the ideas of the Bi-harmonic Modified (BHM) method. THM formulation allowed for the nonlinear terms in the FCH equation to be computed explicitly, leading to fast evaluations at every time step. We investigated the convergence properties of the new approach by using benchmark problems for phase-field models, and we directly compared the performance of the THM method with the recently developed scalar auxiliary variable (SAV) schemes for the FCH equation. The THM modified scheme was able to produce smaller errors than those obtained from the SAV formulation. In addition to this direct comparison with the SAV schemes, we tested the adaptability of our scheme by using an extrapolation technique which allows for errors to be reduced for longer simulation runs. We also investigated the adaptability of the THM method to other 6th order partial differential equations (PDEs) by considering a more complex form of the FCH equation with nonsymmetric double well potential. Finally, we also couple the THM scheme with a higher order time-stepping method, (implicit-explicit) IMEX schemes, to demonstrate the robustness and adaptability of the new scheme. Numerical experiments are presented to investigate the performance of the new approach.

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