- Date: Friday April 16
- Time: 2:00pm
- Venue: Burnside 1205
- You are allowed to bring one (double-sided) sheet of hand-written notes to the exam.

**Initial value problems.**
The main reference to supplement the class notes is the relevant chapters of [Iserles] or [LeVeque], and one of the survey articles of Ernst Hairer (specifically, the first article under 2003 and the one under 2002).
[Flaherty] and [Süli] offer a nice alternative to the above two books.
For a general overview of the subject, have a look at this Scholarpedia article by Lawrence Shampine and Skip Thompson.
For a deeper study, one could start with the three books on Numerical Analysis of Differential Equations coauthored by Ernst Hairer, Christian Lubich, Syvert Nørsett and Gerhard Wanner.

- General concepts: Consistency, local truncation error, order of consistency, convergence, order of convergence, explicit and implicit methods, adaptivity, stiffness
- Stability concepts: Zero stability, absolute (or linear) stability, A-stability, A(α)-stability, L-stability, B-stability, algebraic stability
- Runge-Kutta family: Order conditions, convergence, embedded pairs, stability function, collocation based implicit RK methods: Gauss, Radau, and Lobatto, implementation of implicit RK methods
- Linear multistep methods: Order conditions, root condition, convergence, stability region, root locus curve, Adams and Bashforth families, predictor-corrector pairs, Backward Differentiation Formulae, Dahlquist's barriers, Dahlquist's equivalence theorem
- Geometric integrators: Numerical flow, linear and quadratic invariants, adjoint and composition of numerical methods, symmetric methods, reversible methods

- Finite difference methods: Consistency, convergence, stability, stability in uniform norm, discrete Green's function, stability in 2-norm, conditioning of the discretized system, spectral collocation (or pseudospectral) method
- Finite element methods: Weak formulation, Galerkin approximation, Galerkin orthogonality, Cea's lemma, convergence of FEM, boundary conditions, adaptivity, multigrid idea
- Spectral methods: Fourier series, pointwise convergence, convergence in L2-norm, relation between convergence rate and smoothness, Fourier-Galerkin methods, trapezoidal quadrature and discrete Fourier transform, Gibbs phenomenon, Chebyshev polynomials, Chebyshev-Galerkin methods

- Generalities: Method of lines, Lax–Richtmyer theory, Von Neumann stability analysis, Courant–Friedrichs–Lewy condition, domain of dependence, transport, dissipation, dispersion
- Specific schemes: Euler, leap–frog, Crank–Nicolson, Lax–Friedrichs, Lax–Wendroff, upwind