A Sharp-Interface Active Penalty Method for the Incompressible Navier–Stokes Equations
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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.
KeywordsActive penalty method Sharp mask function Immersed boundary Incompressible flow Navier–Stokes Heat equation
Mathematics Subject Classification65N06 65N35 65N85 76M20 76M22
The authors would like to thank Kirill Shmakov and Geneviève Bourgeois for additional preliminary computations not currently presented. The authors have also greatly benefited from conversations with Dmitry Kolomenskiy, Kai Schneider, Ruben Rosales and Tsogtgerel Gantumur. This work was supported by an NSERC Discovery Grant and the NSERC DAS.
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