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. Existence and uniquenes of solutions of ODEs. Solution of simple ODEs and simple (linear) PDEs. Fourier series and transform. Norms and function spaces. | |
| Numerical ODEs | Runge-Kutta and multistep-methods, convergence of the Euler method, order conditions and stability. Note on ODEs, exercise set 8 from the course of Numerical Mathematics (TMA4215). |
Timetable
| Week | Date | JCS and SM | N | Subject | Read on your own | Some relevan exam questions |
|---|---|---|---|---|---|---|
| 2 | 10.01, 13.01 | ch. 1-2,3, 4.1-4.2 | Introduction to the course. Difference operators and difference formulae. Classification of linear PDEs. Methods for parabolic problems. Preparation to the first two assignments: Boundary value problems. Discretization of the heat equation. | 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 | 17.01, 20.01 | Supervision of the first two compulsory assignments (Nullrommet) | ||||
| 4 | 24.01, 27.01 | 4.1-4.5, 6 | Forward Euler, Backward Euler and Crank-Nicholson. LTE of the theta-method. Method of lines. Methods for elliptic equations. Preparation to the third assignment: Discretization of the Laplace equation in 2D. | Problem 3a) exam June 2010. Exercise 1, June 2007. | ||
| 5 | 31.01, 03.02 | 6, 7.1, 7.2 | Methods for elliptic equations. Methods for advection equations | Exercise 1, exam June 2010. Problem 2, August 2014. Problem 2, June 2014. | ||
| 6 | 07.02, | 7.1–7.4. 5.3 | Methods for advection equations and hyperbolic systems. Preparation to the fourth assignment: Hyperbolic equations. | Problem 4 August 2014. Problem 3 June 2012. | ||
| 7 | 14.02, 17.02 | 7.5–7.6, 5.1-5.4 5.6, 5.8 | CFL condition. (Ch. 7.5 is self-study). Convergence of the Euler method, is self-study (ch 5.4). | Problem 4 Exam May 2013. Ch. 5.4 note. Example page 61-62 note. Problem 3, Exam May 2013. | ||
| 8 | 21.02, 24.02 | 5.9, 7.6, 7.7 | Stability of parabolic problems. Matrix stability. 5.5 is not part of the curriculum. 5.7 is not part of the curriculum. Lax equivalence theorem. Von Neumann stability. | 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 Pb 2 August 2014. | ||
| 9 | 28.02, 03.03 | 6 | Elliptic equations maximum principle. Project work. | Pb 4 May 2013. Pb 3 June 2013 | ||
| 10 | 07.03, 10.03 | Numerical linear algebra. | Pb 2c Aug 2013, Pb 2 May 2013, Pb 1c Aug 2014 | |||
| 11 | 14.03, 17.03 | 13 JCS p 339-349 p 354-356. 14 JCS p 373-387 p 390-391 | Numerical linear algebra. Finite element method. | |||
| 12 | 21.03, 22.03, 24.03 | Project work. | ||||
| 13 | 28.03, 29.03, 31.03 | Project presentation and project work. | Pb 2c Aug 2013, Pb 2 May 2013, Pb 1c Aug 2014 | |||
| 04.04, 18.04 | Easter and excursion | |||||
| 16 | 21.04 | SM chaper 14 p 385-399. See material in It's learning. | Finite element method: Rayleigh–Ritz and Galerkin principles and methods. | Pb 1: May 2009, May 2011, May 2013, Aug 2013, June 2014, Aug. 2014. | ||
| 17 | 25.04, 28.04 | 14 SM p 385-399 | Finite element method. Error estimastes (from Suli and Mayers). 2D Poisson equation. | Pb2 June 2010, Pb2 Aug 2011, Oppg 2 Aug 2012, Oppg 4 June 2012. |