Modeling, Analysis, and Computation of interesting scientific / technological / industrial problems

commonly known as

Prof. Vasilios ALEXIADES Ayres 213 974-4922 alexiades@utk.edu

8-10 Lab/Homework assignments: 40% , Project assignments: 40% , Term/Team Project: 20%

Contact the Office of Disability Services (2227 Dunford Hall, 974-6087) to coordinate reasonable accommodations for documented disabilities.

Involves: scientific problem → math problem → computational algorithm → numerical solution → implications for original scientific problem.

It has become the 3rd pillar of Science, complementing Theory and Experiment.

Need to be solved numerically (approximately), so need

These aims often play against each other, so trade-offs need to be made...

The course will simulate the core aspects of
** Computational Science** including:

writing reports, writing proposals, collaborating with colleagues on a research project, and presenting your work.

I.Crystal precipitation- physical model leading to ODE system - about ODEs - well posedness of IVP - equilibria - root finding (Newton method) - plotting - analysis of the model - Euler scheme - computational errors - consistency-stability-convergence - implementation - classical RK4 and other numerical schemes II.Air pollution: Advection and Diffusion Processes- linear advection - wave propagation - 1st order PDEs - method of characteristics - the general consrvation law u_{t}+ div F = 0 - derivation from first principles - conservation of species - advective and diffusive fluxes - continuity equation - finite volume discretization of u_{t}+ F_{x}= 0 - explicit/implicit - advection ( F = uV ) - explicit upwind scheme - CFL condition - implementation - diffusion ( F = −Du_{x}) - parabolic PDEs - boundary conditions - explicit scheme - CFL condition - advection-diffusion ( F = uV − Du_{x}) - explicit scheme - CFL condition - effect of small/large Peclet number - super-time-stepping acceleration - a few words about Lax-Wendroff and other schemes III.Chemical reactionsvia mass action kinetics IV.Uncertainty Quantification and parameter estimationSome other possible topics: V.Melting and Freezing- phase-change basics, moving boundary problems - Stefan Problem, exact solution, analytic approximations - enthalpy formulation, explicit scheme VI.The catalytic converter- diffusion-reaction model - control problem - calculus of variations - Euler-Lagrange equation - numerical scheme for the forward model VII.Electron beam lithography(inverse problems) - forward scattering (dose to exposure) - inverse problem (exposure to dose) - ill posed problem - Fourier-Poisson integral solution of diffusion equation - Fourier series solution of diffusion equation - Fourier series approximation of the inverse problem - Discrete Fourier Transform, FFT

------------