Abstract

This presentation centers on numerically solving the time-fractional Allen-Cahn equation by substituting the time differential operator with a Caputo-type differential operator. Addressing the challenge of ensuring a solution that adheres to the maximum principle while leaving the dissipation of the energy functional uncertain, the talk employs the method and convex splitting methods for numerical discretization.

In the first part, I mainly introduce a novel dissipative functional designed to explore the dissipative property of the classical energy functional along the time-fractional Allen-Cahn equation, which is still elusive to answer. I will introduce the formation of the Caputo derivative operator and utilize the scheme for discretization. Also, I will briefly explain how the convex splitting method is used to handle the nonlinear source term. After, I will present dissipative functionals derived from the Caputo differential operator, showing that the proposed functional unequivocally exhibits the dissipative property through the scheme and the convex splitting mentioned.

The second part delves into error analysis and preserving the maximum norm. Regardless of the convex splitting method, numerical solutions reveal a convergence order of . At the end of the talk, I will underscore the notion that any improvement in convergence order is impossible unless the convex splitting method is replaced. Overall, the presentation contributes to a deeper understanding of the numerical solution properties and convergence rates of the time-fractional Allen-Cahn equation.