Basic Numerical Methods

Upon completing this module, the students will acquire the basics of numerical mathematics and numerical simulation methods. This includes the theoretical understanding of how a computer calculates with finite floating-point numbers and what kind of errors and inaccuracies may arise from these, and how to reduce or control them same. They will be familiar with basic numerical methods for modeling and simulating algebraic models, linear algebra models, and ordinary and partial differential equations. They will be able to estimate the approximation errors of the methods and determine the algorithmic intensity, and will be able to implement these methods themselves.

Floating point arithmetic, rounding errors, cancellation, numerical interpolation (Lagrange, Newton, Splines), Taylor developments, finite differences and their approximation errors, explicit and implicit time integrators, qudrature, direct and iterative algorithms for matrix inversion, matrix decomposition (LU), solution for the Poisson equation.

Topic Prerequisites

Working knowledge of computer programming in any language (e.g. Matlab, Python, Java), and solid undergraduate knowledge in calculus and vector analysis.

Program / Module

M.Sc. Computational Modeling and Simulation
Module: CMS-COR-NUM - Basic Numerical Methods

Format

2 SWS lecture, 2 SWS exercise, self-study
5 credits

Registration to the course

For students of the Master program "Computational Modeling and Simulation: via CampusNet SELMA

Teachers

Lecture: Dr. Nandu Gopan, Prof. Ivo F. Sbalzarini
Exercises: Mr. Kiran Vinayan

Instruction language: ENGLISH

Script

Lecture notes are available as PDF here.

Time/Place

Winter Term 2024/25

Lecture: Mondays, 5. DS (14:50-16:20) - APB/E023 FIRST LECTURE: OCT 14, 2024
Exercises: Thursdays, 3. DS (11:10-12:40) - FOE/0244/H FIRST TUTORIAL: OCT 17, 2024

LECTURES AND EXERCISES WILL BE IN PRESENCE FOR THE WHOLE SEMESTER, BUT SUPPORTED WITH ONLINE VIDEO RECORDINGS

Link to the lecture videos in OPAL: https://bildungsportal.sachsen.de/opal/auth/RepositoryEntry/32365445133

Exam

Please refer to the OPAL page of the course for exam related information.

Lecture syllabus/Handouts

Oct 09, 2023: Lecture 0 - Introduction to the Course and Organization
Oct 16, 2023: Lecture 1 - Finite-precision arithmetics, IEEE number representation, roundoff and extinction, error propagation, condition numbers, backward error analysis
Oct 23, 2023: Lecture 2 - linear systems of equations, LU decomposition of matrices, Gaussian elimination, iterative linear solvers, Jacobi method
Oct 30, 2023: Lecture 3 - Gauss-Seidel method, SOR method, Conjugate gradient methods, preconditioning schemes
Nov 6, 2023: Lecture 4 - Least-Squares methods, QR decomposition, singular value decomposition
Nov 13, 2023: Lecture 5 - Non-linear least squares, non-linear equations, Newton method, bisection method, secant method
Nov 20, 2023: Lecture 6 - Non-linear systems of equations, quasi-Newton method, rank-1 update, Broyden algorithm, Lagrange interpolation, barycentric interpolation
Nov 27, 2023: Lecture 7 - Interpolation algorithms: Aitken-Neville algorithm, Hermite and Spline interpolation
Optional: Trigonometric interpolation: Discrete Fourier transform and fast Fourier transform algorithms
Dec 4, 2023: Lecture 8 - Numerical integration (quadrature): trapezoidal rule, Simpson rule, Romberg extrapolation, Gauss quadrature
Dec 11, 2023: Lecture 9 - Numerical differentiation: finite difference methods, Romberg extrapolation, Initial value problems of ordinary differential equations, the explicit Euler scheme
Dec 18,2023: Lecture 10 - second-order methods, Heun's method, Runge-Kutta methods, variable step size control, embedded Runge-Kutta, Richardson extrapolation
Jan 8, 2024: Lecture 11 - implicit methods, multistep methods, Systems of ODEs, higher-order ODEs
Jan 15, 2024: Lecture 12 - numerical stability, stiff problems, Lax equivalence theorem, introduction to PDEs
Jan 22, 2024: Lecture 13 - Partial differential equations: parabolic problems, elliptic problems, hyperbolic problems, method of lines, stencil methods, method of characteristics, Richardson method, Crank-Nicholson method, Courant-Friedrichs-Lewy condition
Jan 29, 2024: No Lecture