Numerical methods for differential equations

Learning Outcomes

  1. discretize ordinary differential equations with initial condition using a finite difference method of required precision
  2. discretize ordinary differential equations with boundary conditions using finite difference and finite element methods
  3. discretize elliptic partial differential equations using finite difference and finite element methods
  4. discretize transport equations by corresponding methods
  5. implement numerical schemes on computer using appropriate software and numerically solve them
  6. decide on appropriateness of a numerical scheme for certain types of problems
  7. carry out simulation tests and interpret observations
  8. differentiate between numerical artefacts of inappropriate numerical schemes and physical phenomena in systems under consideration

Forms of Teaching

Lectures

Lectures will be held once per week in total duration of 3 hours. They will be focused on theoretical foundations of numerical methods for differential equations.

Exercises

Auditory exercises will be held once per week in total duration of one hour. Based on specific exercises the understanding of theoretical concepts from lectures is adopted and deepened.

Independent assignments

Students are given homework assignments. Homework points make 40% of the total number of points.

Laboratory

Laboratory exercises are held over 10 weeks of semester (one hour each) and in those weeks together with auditory exercises. In laboratory exercises, the emphasis is on the implementation of numerical methods and they serve as preparation for homework assignments.

Grading Method

Continuous Assessment Exam
Type Threshold Percent of Grade Threshold Percent of Grade
Homeworks 0 % 40 % 0 % 40 %
2. Mid Term Exam: Written 0 % 30 % 0 %
Final Exam: Written 0 % 30 %
Exam: Written 0 % 60 %

Week by Week Schedule

  1. Introduction lecture: motivation examples. Finite difference approximations: derivation, truncation error.
  2. One-step methods for initial value problems: stability, error estimates, convergence.
  3. Multistep methods for initial value problems: stability, error estimates, convergence.
  4. Stiff systems.
  5. Finite difference methods for elliptic equations: Poisson equation. Iterative methods. Multigrid.
  6. Finite difference methods for parabolic equations: the heat equation. Crank-Nicolson method.
  7. Finite difference methods for transport equations: upwind scheme. CFL condition.
  8. Midterm exam
  9. Elements of functional analysis: Sobolev apaces and weak formulation.
  10. Galerkin finite element method for elliptic equations. Helmholtz equation.
  11. Finite element method for reaction - diffusion equations. Petrov-Galerkin method.
  12. Fluid flow simulations. Navier-Stokes equations.
  13. Traffic flow simulation. Discontinuous Galerkin finite element method.
  14. Maxwell equations. Boundary element method.
  15. Final exam

Study Programmes

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Literature

Randall J. LeVeque (2007.), Finite Difference Methods for Ordinary and Partial Differential Equations, SIAM
Alfio Quarteroni (2017.), Numerical Models for Differential Problems, Springer
Alfio Quarteroni, Alberto Valli (2009.), Numerical Approximation of Partial Differential Equations, Springer Science & Business Media
Thomas Rylander, Pär Ingelström, Anders Bondeson (2012.), Computational Electromagnetics, Springer Science & Business Media

For students

General

ID 222537
  Summer semester
5 ECTS
L3 English Level
L1 e-Learning
45 Lectures
15 Exercises
10 Laboratory exercises

Grading System

85 Excellent
70 Very Good
55 Good
45 Acceptable