Adaptive linear second-order energy stable schemes for time-fractional Allen-Cahn equation with volume constraint

TL;DR

This paper introduces adaptive, energy-stable numerical schemes for a time-fractional Allen-Cahn equation with volume constraint, ensuring volume preservation and efficiency in multi-scale simulations.

Contribution

The paper develops novel adaptive linear second-order energy stable schemes combining invariant energy quadratization and scalar auxiliary variable methods for the fractional Allen-Cahn equation with volume constraint.

Findings

Schemes are volume-preserving and unconditionally energy stable on nonuniform meshes.

Algorithms efficiently handle initial singularity and fast dynamics in simulations.

Numerical results demonstrate computational efficiency and accuracy.

Abstract

A time-fractional Allen-Cahn equation with volume constraint is first proposed by introducing a nonlocal time-dependent Lagrange multiplier. Adaptive linear second-order energy stable schemes are developed for the proposed model by combining invariant energy quadratization and scalar auxiliary variable approaches with the recent L1$^{+}$ formula. The new developed methods are proved to be volume-preserving and unconditionally energy stable on arbitrary nonuniform time meshes. The accelerated algorithm and adaptive time strategy are employed in numerical implement. Numerical results show that the proposed algorithms are computationally efficient in multi-scale simulations, and appropriate for accurately resolving the intrinsically initial singularity of solution and for efficiently capturing the fast dynamics away initial time.