The course will simulate the core aspects of
Computational Science including:
•
modeling and computational simulation of physical phenomena
•
writing reports
•
writing proposals
•
collaborating with colleagues
on a Team/Term project
•
and presenting your work.
Start thinking about a Term project topic right away!
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
Some 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
------------ Some comments from happy students -------------