A Doubly Adaptive Penalty Method for the Navier Stokes Equations

TL;DR

This paper introduces an adaptive penalty parameter method for Navier-Stokes equations that improves velocity approximation stability and accuracy by dynamically adjusting the penalty parameter and time-step.

Contribution

It presents a novel adaptive penalty parameter technique with stability analysis and demonstrates its effectiveness through numerical tests, enhancing velocity approximation in Navier-Stokes simulations.

Findings

Adaptive penalty parameter improves velocity stability.

Dynamic adjustment of $\

The method yields good velocity approximations and flexible time-step control.

Abstract

We develop, analyze and test adaptive penalty parameter methods. We prove unconditional stability for velocity when adapting the penalty parameter, $\epsilon,$ and stability of the velocity time derivative under a condition on the change of the penalty parameter, $\epsilon(t_{n+1})-\epsilon(t_n)$. The analysis and tests show that adapting $\epsilon(t_{n+1})$ in response to $\nabla\cdot u(t_n)$ removes the problem of picking $\epsilon$ and yields good approximations for the velocity. We provide error analysis and numerical tests to support these results. We supplement the adaptive-$\epsilon$ method by also adapting the time-step. The penalty parameter $\epsilon$ and time-step are adapted independently. We further compare first, second and variable order time-step algorithms. Accurate recovery of pressure remains an open problem.