Below I present a list that is meant to give an idea of what topics you should expect on the exam.
If you want to know whether a particular topic will be covered, drop me a note or ask me in person.
I also attempted to come up with references to literature that might be useful, either in preparing for the exam or in studying further.
It goes without saying that for any given topic, it is a good idea to check out its Wikipedia page.
- 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.
The main references are Chapters 8, 9, 10, 13 of [Iserles] and Chapters 2, 3, 4 of [LeVeque].
Nice free alternatives are [Flaherty] and [Süli] on finite elements,
and [Flaherty] and [Süli] on numerical solution of PDE's in general, the latter two also including time dependent PDE's.
A good introduction to spectral methods is [Trefethen],
and to multigrid is [Briggs–Henson–McCormick].
For a serious study in the mathematical theory of finite element methods, I would strongly recommend [Brenner–Scott].
- 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
Time dependent PDE.
The main references are Chapters 16, 17 of [Iserles] and Chapters 9, 10 of [LeVeque].
Joseph Flaherty's lecture notes are a good alternative.
There is an earlier book by Randall LeVeque on numerical methods for nonlinear conservation laws,
which is a genuine introduction to the subject.
For spectral methods for time dependent problems this article by Sigal Gottlieb gives a nice overview,
and for further study let me mention [Fornberg] and
- 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