Efficient Convex Optimization-Based Bound-Preserving High-Order Accurate Limiter for Cahn-Hilliard-Navier-Stokes System

Core Concepts

A simple and efficient convex optimization-based post-processing procedure can enforce bounds, conservation, and high-order accuracy for numerical schemes solving the Cahn-Hilliard-Navier-Stokes system.

Abstract

The content discusses a simple and efficient approach to enforce the bound-preserving property of high-order accurate numerical schemes for phase field models, without destroying conservation and accuracy.
Key highlights:

Many numerical methods, especially high-order accurate schemes, do not preserve bounds, which is critical for physical meaningfulness and robustness of numerical computation.
The authors propose a convex optimization-based post-processing procedure to enforce bounds, conservation, and high-order accuracy for numerical solutions of the Cahn-Hilliard-Navier-Stokes (CHNS) system.
The optimization problem is formulated as a nonsmooth convex minimization, which can be efficiently solved using the generalized Douglas-Rachford splitting method with optimal algorithm parameters.
The authors analyze the asymptotic linear convergence rate of the Douglas-Rachford splitting and provide a simple formula to choose nearly optimal algorithm parameters based on the number of out-of-bounds cells.
Numerical tests on a 3D CHNS system demonstrate that the proposed limiter is high-order accurate, very efficient, and well-suited for large-scale simulations, taking at most 20 iterations to converge.

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Key Insights Distilled From

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

by Chen Liu,Bea... at arxiv.org 04-01-2024

https://arxiv.org/pdf/2307.09726.pdf

Deeper Inquiries

How can the proposed bound-preserving limiter be extended to other types of PDEs beyond the Cahn-Hilliard-Navier-Stokes system

The proposed bound-preserving limiter based on convex optimization can be extended to other types of PDEs by adapting the formulation of the minimization problem and constraints to suit the specific characteristics of the new equations. For different PDEs, the cell averages and bounds may vary, requiring adjustments in the objective function and constraints to ensure conservation, accuracy, and bound-preservation. By modifying the cell averages and applying the Zhang-Shu limiter, the approach can be tailored to address the requirements of various PDEs, such as hyperbolic or parabolic equations, diffusion-dominated systems, or other phase field models. The key lies in formulating the optimization problem to enforce the necessary properties while maintaining efficiency and accuracy in the numerical solution.

What are the potential limitations or drawbacks of the convex optimization-based approach compared to other bound-preserving techniques, such as flux limiters or exponential time differencing methods

While the convex optimization-based approach for bound-preserving limiters offers advantages in terms of simplicity, efficiency, and high-order accuracy, there are potential limitations compared to other techniques like flux limiters or exponential time differencing methods. One drawback is the computational cost associated with solving the convex minimization problem in each time step, especially for large-scale simulations with a high number of cells. The complexity of the optimization process may impact the overall efficiency of the numerical scheme, particularly in cases where the number of out-of-bounds cells is significant. Additionally, the optimal algorithm parameters derived from the asymptotic convergence rate analysis may not always translate directly to practical implementations, requiring further tuning and adjustments based on specific problem instances. Furthermore, the generalizability of the convex optimization approach to different types of PDEs may require additional modifications and considerations to ensure effectiveness and robustness across various scenarios.

What are the deeper connections between the asymptotic convergence rate analysis of the Douglas-Rachford splitting and the underlying physics or numerical properties of the Cahn-Hilliard-Navier-Stokes system

The asymptotic convergence rate analysis of the Douglas-Rachford splitting method provides insights into the numerical behavior and efficiency of the bound-preserving limiter for the Cahn-Hilliard-Navier-Stokes system. By understanding the rate at which the optimization process converges towards the solution, researchers can optimize the algorithm parameters and improve the computational efficiency of the limiter. The deeper connections between the convergence rate analysis and the underlying physics or numerical properties of the system lie in the balance between conservation, accuracy, and bound-preservation. The analysis helps in determining the trade-offs between enforcing bounds, maintaining conservation laws, and achieving high-order accuracy in the numerical solution. By linking the convergence rate to the system dynamics and numerical discretization, researchers can fine-tune the limiter to strike the optimal balance between these essential aspects, ensuring reliable and efficient simulations of the Cahn-Hilliard-Navier-Stokes equations.