Qin Li, Fall 2017:

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Numerical Methods for Ordinary Differential Equations

- Basic ODE Theory
- Explicit and implicit methods, stability, Runge-Kutta and multistep methods, stiff problems

Finite Difference Methods for Elliptic Partial Differential Equations

- Iterative methods for solving large linear systems

Finite Difference Methods for Parabolic Partial Differential Equations

- Numerical differentiations
- Consistency, stability and convergence
- Multidimensional problems

Finite Difference Methods for Hyperbolic Partial Differential Equations

- Linear hyperbolic equations and their numerical discretizations
- Basic theory for nonlinear hyperbolic equations: shock formation, weak solution and entropy condition, Riemann problem
- Shock capturing methods: Godnov and Roe methods, high resolution methods (TVD and ENO), kinetic and relaxation methods
- Hamilton-Jacobi equations and the level set method for front propagation

Finite Volume Methods for Partial Differential Equations

Spectral Methods for Partial Differential Equations

- Fast Fourier transform
- Fourier spectral method, pseudospectral methods

semester:

Fall

prereqs:

an undergraduate course in PDEs (on the level of Haberman's book on ``Applied Partial Differential Equations'')
an undergraduate course in numerical analysis (on the level of Burden and Faires's book on ``Numerical Analysis'')
knowledge in a programming language (e.g., Fortran, C/C++, or Matlab)

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