Stability and convergence of Strang splitting. Part II: tensorial Allen-Cahn equations

TL;DR

This paper analyzes the stability and convergence of a second-order Strang-splitting method for vector- and matrix-valued Allen-Cahn equations, establishing maximum principles, energy dissipation, and demonstrating efficiency through numerical examples.

Contribution

It provides rigorous proofs of maximum principles and energy dissipation for the splitting method applied to tensorial Allen-Cahn equations, including both vector and matrix cases.

Findings

Maximum principle holds for vector-valued case.

Modified energy functional is dissipative.

Numerical examples confirm theoretical results.

Abstract

We consider the second-order in time Strang-splitting approximation for vector-valued and matrix-valued Allen-Cahn equations. Both the linear propagator and the nonlinear propagator are computed explicitly. For the vector-valued case, we prove the maximum principle and unconditional energy dissipation for a judiciously modified energy functional. The modified energy functional is close to the classical energy up to $\mathcal O(\tau)$ where $\tau$ is the splitting step. For the matrix-valued case, we prove a sharp maximum principle in the matrix Frobenius norm. We show modified energy dissipation under very mild splitting step constraints. We exhibit several numerical examples to show the efficiency of the method as well as the sharpness of the results.