Numerical Treatment of Allen’s Equation Using Semi Implicit Finite Difference Methods

Authors: Younis A. Sabawi1 & Supa B. Ahmed2 & Hoshman Q. Hamad3
1Department of Mathematics, Faculty of Science and Health, Koya University, Koya KOY45, Koya, Iraq
1Mathematics Education Department, Faculty of Education, Tishk International University, Erbil, Iraq
2Mathematics Education Department, Faculty of Education, Tishk International University, Erbil, Iraq
3Department of Mathematics, Faculty of Science and Health, Koya University, Koya KOY45, Koya, Iraqq

Abstract: This paper aims to propose the semi implicit finite difference method for discretizing Cahn-Allen equation. The stability and convergence analysis are proved. It is shown that the suggest scheme is stable for the usage of the Fourier-Von Neumann technique. The accuracy of the proposed method is first order in time and second order in space. A comparison between the numerical and the exact solutions is supported with two examples. Numerical results are shown that there is a good agreement between the approximate solution and exact solution.

Keywords: Allen Equation, Semi Implicit Finite Difference, Stability of Allen Equation

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Doi: 10.23918/eajse.v8i1p90

Published: June 2, 2022

References

Allen, S. M., & Cahn, J. W. (1979). A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metallurgica, 27(6), 1085-1095.

Boscarino, S., Filbet, F., & Russo, G. (2016). High order semi-implicit schemes for time dependent partial differential equations. Journal of Scientific Computing, 68(3), 975-1001.

Bulut, H. (2017, January). Application of the modified exponential function method to the Cahn-Allen equation. In AIP Conference Proceedings (Vol. 1798, No. 1, p. 020033). AIP Publishing LLC.

Dhumal, M. L., & Kiwne, S. B. (2018). Numerical treatment of fisher’s equation using finite difference method. Bulletin of Pure & Applied Sciences-Mathematics and Statistics, 37(1), 94-100.

Huang, P., & Abduwali, A. (2011). A numerical method for solving Allen-Cahn Equation. Journal of Applied Mathematics & Informatics, 29(5_6), 1477-1487.

Hussein, Y. A. (2011). Combination between single diagonal implicit and explicit runge kutta (sdimex- rk) methods for solving stiff differential equations. Tikrit Journal of Pure Science, 16(1).

Manaa, S. A., Moheemmeed, M. A., & Hussien, Y. A. (2010). A numerical solution for sine-gordon type system. Tikrit Journal of Pure Science, 15(3).

Munguia, M., & Bhatta, D. (2015). Use of cubic b-spline in approximating solutions of boundary value problems. Applications and Applied Mathematics: An International Journal (AAM), 10(2), 7.

Pirdawood, M. A., & Sabawi, Y. A. (2021, September). High-order solution of Generalized Burgers–Fisher Equation using compact finite difference and DIRK methods. In Journal of Physics: Conference Series (Vol. 1999, No. 1, p. 012088). IOP Publishing.

Sabawi, Y. A. (2017). Adaptive discontinuous galerkin methods for interface problems (Unpub- lished doctoral dissertation). PhD Thesis.

Sabawi, Y. A. (2019). A posteriori L_∞ (H^1 ) error bound in finite element approximation of semdiscrete semilinear parabolic problems. In 2019 first international conference of computer and applied sciences (cas) (pp. 102–106).

Sabawi, Y. A. (2020). A posteriori error analysis in finite element approximation for fully discrete semilinear parabolic problems. In Finite element methods and their applications. IntechOpen.

Sabawi, Y. A. (2021). A Posteriori L_∞ (L_2 )+L_2 (H^1 )–Error bounds in discontinuous galerkin methods for semidiscrete semilinear parabolic interface problems. Baghdad Science Journal, 18(3).

Sabawi, Y. A. (2021). Posteriori error bound for fullydiscrete semilinear parabolic integro-differential equations. In Journal of physics: Conference series (Vol. 1999, p. 012085).

Sabawi, Y. A., Pirdawood, M. A., & Khalaf, A. D. (2021). Semi-implicit and explicit runge kutta methods for stiff ordinary differential equations. In Journal of physics: Conference series 1999, 012100).

Sabawi, Y. A., Pirdawood, M. A., & Rasool, H. M. (2021). Model reduction and implicit-explicit runge-kutta methods for nonlinear stiff initial-value problems. In Seventh international scientific conference Iraqi al khwarizmi society. Mosul, Iraq.

Sabawi, Y. A., Pirdawood, M. A., & Sadeeq, M. I. (2021). A compact fourth-order implicit-explicit runge-kutta type method for solving diffusive lotka–volterra system. In Journal of physics: Conference series, 1999, 012103).

Villarreal, J. M. (2020). Approximate solutions to the Allen-Cahn equation using the finite difference method.

Yang, X., Ge, Y., & Zhang, L. (2019). A class of high-order compact difference schemes for solving the Burgers’ equations. Applied Mathematics and Computation, 358, 394-417

Yokus, A., & Bulut, H. (2019). On the numerical investigations to the Cahn-Allen equation by using finite difference method. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 9(1), 18-23.