Trigonometric B-Spline Collocation Method for Solving PHI-Four and Allen–Cahn Equations

TLDR: In this paper, a non-polynomial B-spline interpolant and a weighted scheme for space and time discretization is proposed for solving PHI-Four and Allen-Cahn equations.

Abstract: In this paper, we develop a numerical solution based on nonpolynomial B-spline (trigonometric B-spline) collocation method for solving time-dependent equations involving PHI-Four and Allen–Cahn equations. A three-time-level implicit algorithm has been derived. This algorithm combines the trigonometric B-spline interpolant and the \(\theta \)-weighted scheme for space and time discretization, respectively. Convergence analysis is discussed and the accuracy of the presented method is \(O( {\tau ^{2}+h^{2}} ).\) Applying von Neumann stability analysis, the proposed technique is shown to be unconditionally stable. Three test problems are demonstrated to reveal that our method is reliable, efficient and very encouraging.