A simple and efficient convex optimization based bound-preserving high order accurate limiter for Cahn-Hilliard-Navier-Stokes system

TL;DR

This paper introduces a convex optimization-based limiter for the Cahn-Hilliard-Navier-Stokes system that is high-order accurate, efficient, and suitable for large-scale 3D simulations, ensuring bound-preserving properties without sacrificing accuracy.

Contribution

It develops a simple, optimal parameter selection method for a convex optimization-based limiter, enhancing efficiency and accuracy in bound-preserving schemes for complex PDE systems.

Findings

Limiter achieves high-order accuracy in 3D simulations.

Enforces bounds within 20 iterations per time step.

Computational cost scales linearly with the number of cells.

Abstract

For time-dependent PDEs, the numerical schemes can be rendered bound-preserving without losing conservation and accuracy, by a post processing procedure of solving a constrained minimization in each time step. Such a constrained optimization can be formulated as a nonsmooth convex minimization, which can be efficiently solved by first order optimization methods, if using the optimal algorithm parameters. By analyzing the asymptotic linear convergence rate of the generalized Douglas-Rachford splitting method, optimal algorithm parameters can be approximately expressed as a simple function of the number of out-of-bounds cells. We demonstrate the efficiency of this simple choice of algorithm parameters by applying such a limiter to cell averages of a discontinuous Galerkin scheme solving phase field equations for 3D demanding problems. Numerical tests on a sophisticated 3D Cahn-Hilliard-Navier-Stokes system indicate that the limiter is high order accurate, very efficient, and well-suited for large-scale simulations. For each time step, it takes at most $20$ iterations for the Douglas-Rachford splitting to enforce bounds and conservation up to the round-off error, for which the computational cost is at most $80N$ with $N$ being the total number of cells.