INVESTIGATION OF DIFFERENCE SCHEMES FOR NUMERICAL SOLUTION OF THE ONE-DIMENSIONAL ALLEN-CAHN EQUATION

Mariya Orlova; Andrey Gorunov

doi:doi:10.55421/3034-4689_2026_29_6_119

Home / Journals / Herald of Technological University / Volume 29 Issue 6 / INVESTIGATION OF DIFFERENCE SCHEMES FOR NUMERICAL SOLUTION OF THE ONE-DIMENSIONAL ALLEN-CAHN EQUATION

INVESTIGATION OF DIFFERENCE SCHEMES FOR NUMERICAL SOLUTION OF THE ONE-DIMENSIONAL ALLEN-CAHN EQUATION

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INVESTIGATION OF DIFFERENCE SCHEMES FOR NUMERICAL SOLUTION OF THE ONE-DIMENSIONAL ALLEN-CAHN EQUATION

Journal: HERALD OF TECHNOLOGICAL UNIVERSITY Volume 29 № 6 , 2026

Rubrics: 3. INFORMATION TEORY, COMPUTER TECHNOLOGY AND CONTROL

Mariya Orlova ¹

Andrey Gorunov ²

Author and publication information

Authors:

1. KNITU-KAI im. A. N. Tupoleva (Kafedra Lazernyh i additivnyh tehnologiy, Assistent)
graduate student from 01.01.2023 until now

Kazan, Russian Federation

2. Kazan National Research Technical University named after A.N. Tupolev (Kafedra lazernyh i additivnyh tehnologiy, Professor)
employee

Type:

Article

DOI:

https://doi.org/10.55421/3034-4689_2026_29_6_119

EDN:

https://elibrary.ru/FXUKXG

Pages:

from 119 to 124

Status:

Published

Received:

06.05.2026

Accepted:

31.05.2026

Published:

02.07.2026

Language:

Russian

Keywords:

MATHEMATICAL MODELING, PHASE FIELD METHOD, ALLEN-CAHN EQUATION, FINITE DIFFERENCE METHOD, DIFFERENCE SCHEMES, ORDER OF CONVERGENCE

Abstract and keywords

Abstract:
This paper presents a study of the convergence order and computational efficiency of various difference schemes for the numerical solution of the one-dimensional Allen-Cahn equation used in the phase field method for modeling the diffusive interface. The Allen-Cahn equation takes into account the effect of the gradient energy and the double well potential during a phase transition with stable phase states of 0 and 1. The initial condition for the phase field simulates a stepwise phase separation profile without a transition region. To solve the equation, the finite difference method is used using such difference schemes as explicit, implicit, semi-implicit, classical Crank-Nicholson scheme and the stabilized Crank-Nicholson scheme. The run-through method and Newton's method were used to solve the equations. With the numerical solution of the problem, the formation of a diffusion interface is observed with smoothing of the initial phase field profile. The practical convergence orders in time and space for each scheme are determined, which coincide with the theoretical approximation orders. The maximum values of the L1 error rate were about 10-5 in time and 10-4 in space for the largest grids. The best combination of convergence, accuracy, and performance was demonstrated by the stabilized Crank-Nicholson scheme. The effect of the stabilizing parameter on the accuracy of the solution according to the stabilized Crank-Nicholson scheme is investigated. The highest compliance with the reference solution was obtained with the values of the stabilizing parameter in the range from 0,5 to 1. The results obtained make it possible to reasonably select a difference scheme for solving phase field problems, taking into account the requirements for accuracy and computing resources, in particular, when modeling fast-flowing crystallization processes.

Keywords:
MATHEMATICAL MODELING, PHASE FIELD METHOD, ALLEN-CAHN EQUATION, FINITE DIFFERENCE METHOD, DIFFERENCE SCHEMES, ORDER OF CONVERGENCE

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