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