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

TL;DR

The paper addresses the challenge of enforcing bound preservation and mass conservation for high-order Cahn–Hilliard–Navier–Stokes simulations by introducing a post-processing bound-preserving step based on a convex quadratic minimization, solved efficiently via generalized Douglas–Rachford splitting. A two-stage limiter combines this convex minimization with a Zhang–Shu projection to ensure bounds at quadrature points while preserving conservation and accuracy, and a rigorous asymptotic convergence analysis yields a simple, nearly optimal parameter guideline depending on the number of out-of-bounds cells. The method achieves high-order accuracy and robustness in 3D CHNS tests, with at most about 20 iterations per time step and a computational cost of at most 80N, making it well-suited for large-scale simulations. Overall, the approach provides a practical, scalable, and provably efficient bound-preserving limiter for phase-field DG schemes that preserves both physical bounds and mass in difficult CHNS scenarios.

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.

Figures (9)

• Figure 1: The performance of limiting strategy in the accuracy test of applying both limiters \ref{['gDR-average']} and \ref{['zhang-shu']} with mesh resolution $h = 1/2^5$. Left: the percentage of trouble cells at each time step for the cell average limiter \ref{['gDR-average']}. Right: the number of Douglas--Rachford iterations at each time step. For each time step, at most 15 iterations are needed for \ref{['gDR2']}
• Figure 2: Selected snapshots at time steps $2^n$, where $n=3, 5, \cdots, 11$. 3D views of the evolution of order parameter field.
• Figure 3: Left: the average of order parameter at each time step, which shows the conservation is preserved. Middle: the number of Douglas--Rachford iterations at each time step. Right: the asymptotic linear convergence at time step $128$. The predicted rate is the rate given in Theorem \ref{['thm-rate']}.
• Figure 4: The computational domain of the microstructure simulation.
• Figure 5: Selected snapshots at time steps $50$, $100$, $150$, $200$, and $250$. The first and third rows: 3D views of the evolution of the order parameter field. The second and fourth rows: plots of order parameter warped along the plane $\{z=0.5\}$. The top two rows are without limiters and the bottom two rows are with our limiters.

Theorems & Definitions (13)

• Theorem 2.1
• Proof 1
• Lemma 2.2
• Proof 2
• Definition 2.3
• Lemma 2.4
• Proof 3
• Theorem 2.5
• Lemma 2.6
• Proof 4