| Week | Topics |
| 1 |
Introduction Error Analysis & Interval Arithmetic |
| 2 |
Interpolation: Lagrange, error formula, Chebyshev polynomials Interpolation: divided differences, Newton's polynomials |
| 3 |
Hermite interpolation & splines Polynomial Approximation (least-squares optimal) |
| 4 |
Trigonometric Interpolation & Fourier Transform FFT |
| 5 |
Numerical differentiation & Richardson extrapolation Trapezoid, midpoint, Simpson's rules |
| 6 |
General quadratures, error formulas Gaussian quadrature |
| 7 |
Romberg integration / Adaptive methods Intro to ODE/IVP, Euler's method analyzed |
| 8 |
Intro to Runge-Kutta & Linear multistep methods Consistency, stability & order of accuracy |
| 9 |
Taylor series methods, automatic differentiation Stiff ODEs, linear stability of methods |
| 10 |
Numerics for DAEs; continuation methods BVP for ODEs (including intro to FEMs) |
| 11 |
Explicit & implicit finite difference methods for the Heat Equation (consistency, stability, convergence, accuracy, computational cost); Explicit-implicit schemes for reaction-diffusion PDEs |
| 12 |
Intro to Galerkin methods Poisson Equation in 2D: (iterative method & via FFT) |
| 13 |
Poisson Equation in 2D: Finite Element Methods Lagrangian methods for nonlinear PDEs (ray tracing) |
| 14 | Review via Applications |