Parallel energy-stable solver for a coupled Allen-Cahn and Cahn-Hilliard system

TL;DR

This paper introduces a parallel, energy-stable numerical solver for the coupled Allen-Cahn and Cahn-Hilliard equations, featuring adaptive time stepping and domain decomposition for high-performance computing.

Contribution

It develops a novel polynomial approximation for the free energy functional, ensuring unconditional energy stability and scalability on supercomputers.

Findings

Second-order accuracy in space and time

Energy stability with large time steps

Scalability to over ten thousand cores

Abstract

In this paper, we study numerical methods for solving the coupled Allen-Cahn/Cahn-Hilliard system associated with a free energy functional of logarithmic type. To tackle the challenge posed by the special free energy functional, we propose a method to approximate the discrete variational derivatives in polynomial forms, such that the corresponding finite difference scheme is unconditionally energy stable and the energy dissipation law is maintained. To further improve the performance of the algorithm, a modified adaptive time stepping strategy is adopted such that the time step size can be flexibly controlled based on the dynamical evolution of the problem. To achieve high performance on parallel computers, we introduce a domain decomposition based, parallel Newton-Krylov-Schwarz method to solve the nonlinear algebraic system constructed from the discretization at each time step. Numerical experiments show that the proposed algorithm is second-order accurate in both space and time, energy stable with large time steps, and highly scalable to over ten thousands processor cores on the Sunway TaihuLight supercomputer.