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I am currently researching finite element schemes for the approximation of the incompressible Navier-Stokes equations (NSE). Below is a list of topics on which I am currently working or have worked.
• Artificial Compression

• An artificial compression, or artificial compressibility, method is one where the incompressiblity constraint in the continuity equation of the NSE is relaxed appropriately, e.g., the continuity equation is replaced by $$\varepsilon p_t+\nabla\cdot u=0.$$ Doing this allows for the velocity and pressure to be decoupled, allowing for faster numerical methods. My recent work, with W. Layton and V. DeCaria, introduces a second-order, conservative, unconditionally stable artificial compression method based on a stabilized Crank-Nicolson Leapfrog scheme. This method fully decouples the velocity and pressure, allowing for the pressure to be advanced in time explicitly. A link to a preprint of our results is here, and a full reference is below.

[1] V. DeCaria, W. Layton, and M. McLaughlin, A conservative, second order, unconditionally stable artificial compression method, CMAME, (accepted 2017), pp. 1-21.

• Time Filters

• The use of artificial compression methods introduces nonphysical acoustics that manifest as additional errors in the pressure. Time filters, used in geophysical fluid dynamics (GFD), can be used to damp these acoustics. One of the simplest, introduced by A.J. Robert and R. Asselin and called the RA filter out of deference, damps waves by applying a weighted curvature correction, which appears as a discrete second derivative. My recent work with time filters involves applying the RA filter to solutions obtained using the algorithm from [1]. The research, conducted jointly with W. Layton and V. DeCaria, confirms that the RA filter does indeed reduce nonphysical acoustics in the pressure at little computational cost. I am also working with W. Layton, A. Guzel, and Y. Rong on a computational study of the effect that different time filters have on various artificial compression schemes.

[2] V. DeCaria, W. Layton, and M. McLaughlin, An analysis of the Robert-Asselin time filter for the correction of nonphysical acoustics in an artificial compression method, (submitted), pp. 1-21.

[3] A. Guzel, W. Layton, M. McLaughlin, and Y. Rong, A computational study of artificial compression methods with time filtering, (in preparation).

• Ensembles

• Ensemble algorithms, used in weather simulations, are used to deal with uncertainties in given data. I am currently examining fast ensemble algorithms introduced by W. Layton and N. Jiang and am looking to extend them to an artificial compression framework. Furthermore, I am interested in applying artificial compression schemes to natural convection ensemble problems.

• Variable Viscosity

• The NSE with variable viscosity $$\nu(x,t)$$ is considered the simulation of many physical phenomena, e.g., eddy viscosity models of turbulence. Therefore, stable and accurate algorithms incorporating variable viscosity are essential. My work with S. Khankan and V. DeCaria (referenced below) introduces first- and second-order schemes for modeling the NSE with variable viscosity. The report shows that the methods are unconditionally stable, and numerical experiments confirm theoretical error results.

[4] V. DeCaria, S. Khankan, and M. McLaughlin, Time-stepping methods for the Navier-Stokes equations with fluctuating viscosity, (in preparation), pp. 1-18.