Timetable
This schedule is not final but this is the anticipated curriculum of TMA4212.
JCS = John C. Strikwerda's book on finite differences
SM = Suli and Mayers, An introduction to Numerical Analysis
N = course note This note is still under construction (especially chapter 7). Please let me know of any mistakes you find.
Prerequisites
| Subject | Topics | subtopics |
|---|---|---|
| Linear algebra | Basics | Vector and matrix norms; Symmetric Positive Definite matrices; Inner product spaces; Linear independence; basis of a vector space; orthonormal basis. |
| Matrix factorizations | Diagonalization and orthogonal diagonalization of matrices; LU decomposition (Gaussian elimination, pivoting); Cholesky factorization; QR factorization; SVD; Jordan Canonical form; Schur factorization. | |
| Topics of interest in numerical linear algebra | Spectral radius; Gershgoring's theorem; Condition number; Neumann series. | |
| Iterative methods | Newton method; fixed point iteration; convergence of the basic iterative methods (Jacobi, Gauss-Seidel and SOR) for linear systems. | |
| Calculus | Taylor theorem | |
| Solution of ODEs | Numerical methods for ODEs. | Runge-Kutta and multistep, implementation, stability analysis, order conditions, convergence of the Euler method. |
Timetable
| Week | Date | JCS and SM | N | Subject | Read on your own | Some relevan exam questions |
|---|---|---|---|---|---|---|
| 2 | 05.01, 09.01 | ch. 1-2,3 | Introduction to the course. Difference operators and difference formulae. Boundary value problems. | Ch. 1, last part of Ch 3.1.2, beginning of Ch. 3.2, Ch. 3.3. | Problem 1a, 1b, 1c, May 2009. Problem 5, June 2014. | |
| 3 | 12.01, 16.01 | 4.1-4.5 | Classification of linear PDEs. Methods for parabolic problems. Forward Euler, Backward Euler and Crank-Nicholson. LTE of the theta-method. Method of lines. | Problem 3a) exam June 2010. | ||
| 4 | 19.01, 23.01 | 6 | Methods for elliptic equations. | Exercise 1, exam June 2010. Problem 2, August 2014. Problem 2, June 2014. | ||
| 5 | 26.01, 30.01 | 7.1–7.4. | Methods for advection equations and hyperbolic systems. CFL condition. | Problem 4 August 2014. | ||
| 6 | 02.02, 06.02 | 7.5–7.6. 5.1–5.4, 5.6 | Advection equations and hyperbolic systems. Stability of parabolic problems. | Problem 3 June 2012. | ||
| 7 | 09.02, 13.02 | 5.6, 5.8, 5.9 | Matrix stability. Lax equivalence theorem. Von Neumann stability. 5.5 is not part of the curriculum. 5.7 is not part of the curriculum. | Problem 4 Exam May 2013. Ch. 5.4 note. Example page 61-62 note. Problem 3, Exam May 2013. | ||
| 8 | 16.02, 20.02 | 7.6, 7.7 | More on von Neumann stability. Dissipation and dispersion. | Ex.1b May 2004, Ex.3c June 2007, Ex.3c June 2010, Ex.2b May 2011, Ex.1b Aug 2011, Ex.3b Aug 2014, Ex. 3c June 2006 | ||
| 9 | 23.02, 27.02 | 6 | Elliptic equations maximum principle. Project work. | Pb 2 August 2014. Pb 4 May 2013. | ||
| 10 | 02.03, 06.03 | 13 JCS p 339-349 p 354-356 | Numerical solution of linear systems. | Pb 3 June 2013 | ||
| 11 | 09.03, 13.03 | Project work. | ||||
| 12 | 16.03, 20.03 | Project work and presentation. Numerical linear algebra. | ||||
| 13 | 23.03, 27.03 | 14 JCS p 373-387 p 390-391 | Pb 2c Aug 2013, Pb 2 May 2013, Pb 1c Aug 2014 | |||
| 15 | 10.04 | 14 SM p 385-399 | Finite element method: Rayleigh–Ritz and Galerkin principles and methods. | Pb 1: May 2009, May 2011, May 2013, Aug 2013, June 2014, Aug. 2014. | ||
| 16 | 13.04, 17.04 | SM chaper 14 p 385-399. See material in It's learning. | Finite element method. Error estimastes (from Suli and Mayers). 2D Poisson equation. Exam problems. | Pb2 June 2010, Pb2 Aug 2011, Oppg 2 Aug 2012, Oppg 4 June 2012. |