TY - JOUR
T1 - The analysis of an efficient numerical scheme for the Allen–Cahn equations using the Galerkin method
AU - Chin, Pius W.M.
N1 - Funding Information:
The research contained in this article has been supported by Sefako Makgatho Health Sciences University, Medunsa 0204, Ga-rankuwa, Pretoria, South Africa
Publisher Copyright:
© 2021 Elsevier B.V.
PY - 2022/2
Y1 - 2022/2
N2 - In this paper, we propose an efficient numerical scheme for the Allen–Cahn equations. We show theoretically using the Galerkin method and the compactness theorem that the solution of the afore-mentioned equation exists and is unique in appropriate spaces with the interaction length parameter α well controlled. We further, show numerically that the proposed scheme is stable and converge optimally in the L2 as well as the H1-norms with its numerical solution preserving all the qualitative properties of the exact solution. With the help of an example and a carefully chosen α, we use numerical experiments to justify the validity of the proposed scheme.
AB - In this paper, we propose an efficient numerical scheme for the Allen–Cahn equations. We show theoretically using the Galerkin method and the compactness theorem that the solution of the afore-mentioned equation exists and is unique in appropriate spaces with the interaction length parameter α well controlled. We further, show numerically that the proposed scheme is stable and converge optimally in the L2 as well as the H1-norms with its numerical solution preserving all the qualitative properties of the exact solution. With the help of an example and a carefully chosen α, we use numerical experiments to justify the validity of the proposed scheme.
KW - Allen–Cahn equations
KW - Galerkin method
KW - Non-standard finite difference method
KW - Nonlinear equation
KW - Optimal rate of convergence
KW - Stability
UR - http://www.scopus.com/inward/record.url?scp=85117216097&partnerID=8YFLogxK
U2 - 10.1016/j.cnsns.2021.106061
DO - 10.1016/j.cnsns.2021.106061
M3 - Article
AN - SCOPUS:85117216097
SN - 1007-5704
VL - 105
JO - Communications in Nonlinear Science and Numerical Simulation
JF - Communications in Nonlinear Science and Numerical Simulation
M1 - 106061
ER -