Some new computational methods for the Allen‐Cahn equation with non‐periodic boundary conditions arising in computational fluid dynamics

Abstract

Purpose

The purpose of this paper is to present some numerical methods based on different time stepping and space discretization methods for the Allen‐Cahn equation with non‐periodic boundary conditions.

Design/methodology/approach

In space the equation is discretized by the Chebyshev spectral method, while in time the exponential time differencing fourth‐order Runge‐Kutta (ETDRK4) and implicit‐explicit scheme are used. Also, for comparison the finite difference scheme in both space and time is used.

Findings

It is found that the use of implicit‐explicit scheme allows use of a large time‐step, since an explicit method has less order of accuracy as compared to implicit‐explicit method. In time‐stepping the proposed ETDRK4 does not behave well for this special kind of partial differential equation.

Originality/value

The paper presents some numerical methods based on different time stepping and space discretization methods for the Allen‐Cahn equation with non‐periodic boundary conditions.

Citation

Khan, Y., Ali, I., Islam, S. and Biao Wu, Q. (2013), "Some new computational methods for the Allen‐Cahn equation with non‐periodic boundary conditions arising in computational fluid dynamics", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 23 No. 4, pp. 588-597. https://doi.org/10.1108/09615531311323755