Finite element solution of nonlocal Cahn-Hilliard equations with feedback control time step size adaptivity

TL;DR

This paper evaluates feedback control-based adaptive time stepping schemes for the nonlocal Cahn-Hilliard equation, demonstrating improved efficiency and stability, especially with the PC11 predictive controller in 3D simulations.

Contribution

It introduces and compares three feedback control-based adaptive time step schemes for the nonlocal Cahn-Hilliard equation, highlighting the superior performance of the PC11 controller.

Findings

PC11 controller outperforms others in 3D tests

Adaptive schemes reduce total number of time steps

Mass conservation and energy decay are maintained

Abstract

In this study, we evaluate the performance of feedback control-based time step adaptivity schemes for the nonlocal Cahn-Hilliard equation derived from the Ohta-Kawasaki free energy functional. The temporal adaptivity scheme is recast under the linear feedback control theory equipped with an error estimation that extrapolates the solution obtained from an energy-stable, fully implicit time marching scheme. We test three time step controllers with different properties: a simple Integral controller, a complete Proportional-Integral-Derivative controller, and the PC11 predictive controller. We assess the performance of the adaptive schemes for the nonlocal Cahn-Hilliard equation in terms of the number of time steps required for the complete simulation and the computational effort measured by the required number of nonlinear and linear solver iterations. We also present numerical evidence of mass conservation and free energy decay for simulations with the three different time step controllers. The PC11 predictive controller is the best in all three-dimensional test cases.