Efficient linear schemes for a penalized ternary Cahn-Hilliard system

TL;DR

This paper introduces three novel linear numerical schemes for a penalized ternary Cahn-Hilliard system, focusing on accuracy, efficiency, and energy stability, with extensions to multi-component systems and comprehensive simulations.

Contribution

It presents new linear, energy-stable schemes for the ternary Cahn-Hilliard system, including a decoupled first-order, a conditionally stable reduced-cost, and a coupled second-order method.

Findings

The first scheme is unconditionally energy stable and decoupled.

The second scheme reduces computational cost while maintaining energy stability.

Numerical simulations demonstrate the effectiveness and efficiency of the proposed schemes.

Abstract

In this work we introduce novel numerical schemes for a penalized version of the ternary Cahn-Hilliard system for the purpose of creating accurate and efficient numerical schemes of interfacial dynamics with three components as well as some results extending these ideas to systems with four or more components. The first scheme is linear, decoupled, first order accurate, and unconditionally energy stable. Next, we present a second scheme which is a conditionally energy stable modification of the first scheme, but has greatly reduced computational cost. Finally, we present a third scheme which is linear and second order accurate but the unknowns are coupled. Moreover, we present several numerical simulations in two and three dimensions to give a comprehensive overview of each scheme and the cost-benefit analysis associated with designing a method for energy-stability, efficiency, and accuracy.