We will begin with ODE solvers applied to both initial and boundary value problems. Our application will be to find the eigenstates of a quantum mechanical problem or of an optical waveguide.
We will introduce the idea of finite-differencing of differential operators. Our application will be to two problems: vibrating modes of a drum and the evolution of potential vorticity in an advection-diffusion problem of fluid mechanics.
Transform methods for PDEs will be introduced with special emphasis given to the Fast-Fourier Transform. We will revisit the potential vorticity in an advection-diffusion problem of fluid mechanics by using these spectral techniques.
For subtle computational domains, the use of a finite element scheme is compulsory. The steady-state flow of a fluid over various airfoils will be considered.
- (a) Initial value problems
- (b) Euler method, 2nd- and 4th-order Runge-Kutta, Adams-Bashford
- (c) Stability and time stepping issues
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(d) Boundary values problems: shooting/collocation/relaxation
- (2) Finite Difference Schemes for Partial Differential Equations: (3 weeks)
- (a) Collocation
- (b) Stability and CFL conditions
- (c) Time and space stepping routines
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(d) Tri-diagonal matrix operations
- (3) Spectral Methods for Partial Differential Equations: (3 weeks)
- (a) The Fast-Fourier transform (FFT)
- (b) Chebychev transforms
- (c) Time and space stepping routines
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(d) Numerical filtering algorithms
- (4) Finite Element Schemes for Partial Differential Equations: (2 weeks)
- (a) Mesh generation
- (b) Advanced matrix operations
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(c) Boundary conditions