A Sharp-Interface Active Penalty Method for the Incompressible Navier-Stokes Equations

TL;DR

This paper introduces a high-order accurate active penalty method with a sharp interface for solving the incompressible Navier-Stokes equations, improving spatial accuracy while maintaining computational efficiency.

Contribution

It presents a novel active penalty term with a sharp mask function, achieving second and third order convergence in fluid simulations.

Findings

Achieves second and third order convergence in test cases

Maintains similar time step restrictions as traditional penalty methods

Demonstrates effectiveness in 1D and 2D numerical examples

Abstract

The volume penalty method provides a simple, efficient approach for solving the incompressible Navier-Stokes equations in domains with boundaries or in the presence of moving objects. Despite the simplicity, the method is typically limited to first order spatial accuracy. We demonstrate that one may achieve high order accuracy by introducing an active penalty term. One key difference from other works is that we use a sharp, unregularized mask function. We discuss how to construct the active penalty term, and provide numerical examples, in dimensions one and two. We demonstrate second and third order convergence for the heat equation, and second order convergence for the Navier-Stokes equations. In addition, we show that modifying the penalty term does not significantly alter the time step restriction from that of the conventional penalty method.