.. _syllabus:

===========
Syllabus 
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Please note that these schedules are tentative and they may be modified if needed.

**Week 1:**
Motivation and introduction to differential equations and concepts of accuracy and stability.
ODEs: 1-step Euler methods: derivation; explicit; implicit; accuracy and stability; trapezoidal rule.
Methods of derivation: Taylor series; undetermined coefficients; polynomial fitting.



**Week 2:**
Motivation and introduction to differential equations and concepts of accuracy and stability.
ODEs: 1-step Euler methods: derivation; explicit; implicit; accuracy and stability; trapezoidal rule.
Methods of derivation: Taylor series; undetermined coefficients; polynomial fitting.




**Week 3:**
ODEs: Compound methods: Predictor-Corrector; Runge-Kutta.
Extrapolation methods: Richardson; Gragg; Bulirsch-Stoer.
Systems of equations: stiffness.



  
**Week 4:**
ODEs: Two-point boundary-value problems: shooting; relaxation; projection.




**Week 5:**
PDEs: Classification.
General concepts of finite-difference approximations.
Methods for parabolic equations: Forward-Time Centered-Space (FTCS; explicit); Backward-Time Centered-Space (BTCS: implicit); Crank-Nicolson.
Von Neumann stability analysis.



**Week 6:**
PDEs: Methods for hyperbolic equations: Failure of FTCS; Lax method.
Courant-Friedrich-Lewy (CFL) condition.
Other types of errors: phase (dispersion errors); nonlinear (numerical diffusion); transport errors (leading to upwind schemes).


**Week7:**
PDEs: Methods for hyperbolic equations (continued): Second order methods: Leapfrog; 2 step Lax-Wendroff.
Higher order finite-difference methods: 3rd order Essentially Non-Oscillatory; 5th order Weighted ENO (WENO).


**Week 8:**
PDEs: Methods for elliptic equations: 5-point finite-difference stencil.
Review of direct solvers: Gaussian elimination; LU.
Review of iterative solvers: Jacobi iteration; Gauss-Seidel iteration; Successive Over-Relaxation; Conjugate Gradient; preconditioning; accelerators (red-black coloring; multigrid).
Dirichlet/Neumann boundary conditions.




**Week 9:**
Multidimensional problems: Alternating Direction Implicit (ADI).
Other rapid elliptic solvers: Spectral methods, including Fourier; Chebyshev; pseudo-spectral methods; FFTs; dealiasing; difference between Galerkin, tau, collocation methods.



**Week 10:**
A glimpse at other methods (at discretion of instructor): Finite element, finite volume, etc.