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Title of Project: Using integral methods for simulation of fluids and waves

**Faculty member: F.-J. Sayas, M. Depersio, Y. Weng and C. Flores**

Boundary integral equation methods (loosely referred to as the Boundary Element Method by numerical analysts and practitioners) are a powerful set of schemes for simulation of a selected but important group of physical problems of interest, with a focus on exterior problems (equations taking place around obstacles).

This project is about combining a collection of numerical tools, all helping to develop a new simulation method for transient viscous flows in two dimensional settings: (a) time-stepping (discretization in time) will be carried out using Convolution Quadrature techniques, which pretty much work as a black-box for dynamical system whose transfer function is known; (b) space discretization will be developed using a very simple-minded collection of sources and observation points (Dirac deltas) that thin out integral equations to become simple matrices of interactions between particles.

What you would be doing is: (a) understand the MATLAB code we have been developing for the wave equation; (b) understand similarities and differences between the equations of acoustics with the equations of viscous fluids and how these ones affect the integral representations; (c) work out a simple but efficient (vectorized) MATLAB code for the discrete point-source based transfer function that produces the velocity and pressure fields for a viscous flow around one or several obstacles; (d) finally, set up the entire simulation tool by adapting the code for the transfer functions to the time-stepping convolution quadrature tool.

Title of Project: Field interpolation with the reverse heat equation

**Faculty member: L. Rossi and T. Driscoll, M. Hassell, Y. Zheng**

Title of Project: Processing microscopic lipid images

**Faculty member: R. Braun and T. Driscoll, T. Qiu, D. Chapp**

Title of Project: Towards the numerical solution of hyperbolic conservation laws with Chebfun

**Faculty member: T. Driscoll, T. Sanchez-Vizuet, M. Moye**

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