# Efficient and accurate numerical schemes for a hydrodynamically coupled phase field diblock copolymer model

## TL;DR

This paper introduces efficient, second-order numerical schemes for simulating a complex hydrodynamically coupled diblock copolymer model, ensuring energy stability and computational efficiency through innovative discretization techniques.

## Contribution

The paper develops unconditionally energy stable, linear, second-order schemes using the Invariant Energy Quadratization and projection methods for a coupled copolymer model, improving accuracy and efficiency.

## Key findings

- Schemes are unconditionally energy stable.
- Numerical experiments validate accuracy and stability.
- Systems are linear and efficiently solvable.

## Abstract

In this paper, we consider numerical approximations of a hydrodynamically coupled phase field diblock copolymer model, in which the free energy contains a kinetic potential, a gradient entropy, a Ginzburg-Landau double well potential, and a long range nonlocal type potential. We develop a set of second order time marching schemes for this system using the "Invariant Energy Quadratization" approach for the double well potential, the projection method for the Navier-Stokes equation, and a subtle implicit-explicit treatment for the stress and convective term. The resulting schemes are linear and lead to symmetric positive definite systems at each time step, thus they can be efficiently solved. We further prove that these schemes are unconditionally energy stable. Various numerical experiments are performed to validate the accuracy and energy stability of the proposed schemes.