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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 and Time Adaptivity

• 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. The work below, joint 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.

[1] V. DeCaria, W. Layton, and M. McLaughlin, A conservative, second order, unconditionally stable artificial compression method, Comput. Methods Appl. Mech. Engrg., 2017, pp. 733-747. Preprint

Currently, I am focusing on artificial compression schemes that can be adapted in time. One recently submitted paper, with R.M. Chen and W. Layton, considers a slightly modified artificial compression method that is unconditionally stable and fully adaptable (with respect to the timestep). The work also considers the continuous model corresponding to the scheme, and shows convergence to the incompressible NSE.

[2] R.M. Chen, W. Layton, and M. McLaughlin, Analysis of variable-step/non-autonomous artificial compression methdos, (submitted). Preprint

Further studies on this topic will include more sophisticated variable-step schemes and the examination of Variable-Step, Variable-Order (VSVO) methods.

• Reduced Order Modeling and Data Assimilation

• I am also interested in using the fact that artificial compression methods advance pressure explicitly in time to utilize pressure data in reduced order modeling (ROM) and data assimilation (DA) schemes. Currently, I am working with V. DeCaria, W. Layton, and M. Schneier to introduce algorithms that take advantage of plentiful pressure data.

• Ensembles and Natural Convection

• 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, with J. Fiordilino, have introduced a first order artificial compression scheme to natural convection ensemble problems.

[3] J. Fiordilino and M. McLaughlin, An artificial compression ensemble timestepping algorithm for flow problems, (submitted), pp. 1-20. Preprint

• 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.

[4] 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, accepted, NMPDE, pp. 1-21.

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

• 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.

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