## Computational Fluid Dynamics and Numerical Analysis

### Lagrangian and Semi-Lagrangian for pure advection

A couple of students have been trying to grasp the beauty of Lagrangian and Semi-Lagrangian scheme for pure advection problems which arises in image matching. We have created an online course on the numerical solution of the 1D linear advection equation.

### EPDiff: Euler-Poincaré Equation in the full group of Diffeomorphisms

Shape analysis is predicated on comparison of differences relative to template coordinates in the Grenander deformable template model. Specifically shape is encoded via momentum vactors in template coordinates. These momenta are analogous to solitons, wave fronts and higher dimensional forms found in classical mechanics. Therefore one can apply methods established in computational mechanics.

It is important to develop suitable algorithms for evolution of shapes in Computational Anatomy via EPDiff. Hence the need to develop a numerical method for solving a Hamiltonian that preserves energy as well as linear and angular momentum? But this may not be possible unless adaptive timestep is used according to Ge and Marsden. So can we sacrifice energy for linear and angular momentum or vice-versa.

### Turbulence modelling

Asymptotic analysis of turbulence models can be used as an effective tool in verifying both numerical codes and turbulence models as exemplified by the following:

- K. Gersten (ed.) (1996)
*IUTAM Symposium on Asymptotic methods for Turbulent Shear Flows at High Renolds Numbers,*Kluwer Academic Publishers. - B. Mohammadi and O. Pirroneau (1993)
*Analysis of the k-epsilon turbulence model,*J. Wiley & Sons. - W.R.C. Phillips and J. T. Ratnanather (1990)
*The Outer Region of a Turbulent Boundary Layer,*Physics of Fluids A (Fluid Dynamics), vol.2, no.3, p. 427-434.

We have created an online course on a simple model of turbulent flow in a plane channel.

### Forward-backward parabolic equations

Not only do these equations arise in the numerical solution of separating boundary layers but also in stochastic processes in financial modelling, particle transport problems and modelling of counter-current separator. Examples in thermal boundary layer separation can be found here.

### Convection-diffusion equations with source and sink terms

The k-&epsilon turbulence model presents interesting problems in numerical analysis. First is the problem of low values of k and &epsilon in the freestream just outside of a turbulent boundary layer. Asymptotic analysis (Mohammadi & Pirroneau; Deriat) show that a lower bound for &epsilon in these regimes is necessary to guarantee non-negative solution. The uniqueness of numerical solution of k-&epsilon equations using Newton-Raphson solvers was left unanswered in my thesis. This may be due in part to the presence of nonlinear source and sink terms in the coupled convection-diffusion equations.

### Numerical Solution of Infinite Integrals of Products of Bessel Functions

A problem in auditory physiology requires computing infinite integrals involving products of two Bessel functions of the first or second kind. Such integrals occur in fluid dynamics, elasticity, electrodynamics and biophysics to name but a few applications. We have developed and tested a MATLAB toolbox called IIPBF. It was used to compute kernels in a system of Fredholm Integral equation of the second kind that arose in models of a viscous fluid jet impinging on an infinite plane (Davis et al. 2012; Davis et al. 2013).

### Recommended books

This selection of recent and old books is evidently biased:

- K.W. Morton and D.F. Mayers.
*Numerical Solution of Partial Differential Equations* - K.W. Morton.
*Numerical Solution of Convection-Diffusion Problems* - R.D. Richtmyer and K.W. Morton.
*Difference Methods for Initial Value Problems* - A. Iserles.
*A first course in Numerical Analysis of Differental Equations* - E. Hairer, C. Lubich, G. Wanner.
*Geometric Numerical Integration* - B. Leimkuhler and S. Reich.
*Simulating Hamiltonian Dynamics* - L.N. Trefethen.
*Spectral Methods in MATLAB* - L.N. Trefethen and D. Bau III.
*Numerical Linear Algebra* - E.E. Süli and D.F. Mayers.
*An Introduction to Numerical Analysis* - G.H. Golub and C.F. van Loan.
*Matrix Computations* - Any books by Gilbert Strang!