The course will simulate the core aspects of
Computational Science including:
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modeling and computational simulation of physical phenomena
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writing reports
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writing proposals
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collaborating with colleagues
on a Team/Term project
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and presenting your work.
Start thinking about a Term project topic right away!
... see course page for some links/ideas...
Topics / Content
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 - the general conservation law ut + div F = 0 - derivation from first principles - conservation of species - advective and diffusive fluxes - continuity equation - constitutive laws (for non-advective fluxes) - finite volume discretization of ut + Fx = 0 - explicit/implicit - diffusion ( F = −Dux ) - parabolic PDEs - boundary conditions - explicit scheme - CFL condition - super-time-stepping acceleration - advection ( F = uV ) - explicit upwind scheme - CFL condition - implementation - linear advection - wave propagation - 1st order PDEs - method of characteristics - advection-diffusion ( F = uV − Dux ) - explicit scheme - CFL condition - effect of small/large Peclet number - a few words about Lax-Wendroff and other schemes III. Chemical reactions via mass action kinetics IV. Uncertainty Quantification and parameter estimation ideas V. Neural Networks, Machine Learning, AI ideas Some other possible topics: VI. Melting and Freezing - phase-change basics, moving boundary problems - Stefan Problem, exact solution, analytic approximations - enthalpy formulation, explicit scheme VII. The catalytic converter - diffusion-reaction model - control problem - calculus of variations - Euler-Lagrange equation - numerical scheme for the forward model VIII. 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