A Sharp-Interface Active Penalty Method for the Incompressible Navier-Stokes Equations
This work provides a simple modification to the volume penalty method that significantly improves accuracy for fluid simulations with complex boundaries, benefiting computational fluid dynamics practitioners.
The authors introduce an active penalty method for the incompressible Navier-Stokes equations that achieves second and third order spatial convergence, overcoming the first-order limitation of standard volume penalty methods. Numerical examples in 1D and 2D demonstrate second order convergence for Navier-Stokes and up to third order for the heat equation, without worsening the time step restriction.
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.