# Research

## Artificial Compression Methods

I have been researching artificial compression methods, and have coauthored (with William Layton and Michael McLaughlin) a paper on the subject to be published soon in CMAME. We present a novel time-stepping method for solving the incompressible Navier-Stokes equations which is unconditionally stable, second order accurate in time, and conserves energy. Artificial compression methods have long been used to solve the incompressible NSE numerically by replacing the incompressibility constraint $\nabla \cdot u = 0$ with an equation that relaxes this constraint, most commonly $\varepsilon p_t + \nabla \cdot u = 0$ or $-\varepsilon \Delta p_t + \nabla \cdot u = 0$, with $\varepsilon$ being a small positive constant. This allows the decoupling of velocity and pressure, but introduces nonphysical acoustic waves which can degrade the quality of the solution, which we also address.