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.
|Journal||Communications in Nonlinear Science and Numerical Simulation|
|Publication status||Published - Feb 2022|
- Allen–Cahn equations
- Galerkin method
- Non-standard finite difference method
- Nonlinear equation
- Optimal rate of convergence