Numerical solving of partial differential equations

2022/2023
Programme:
Mathematics, Second Cycle
Year:
1 ali 2 year
Semester:
first or second
Kind:
optional
Group:
M4
ECTS:
6
Language:
slovenian, english
Hours per week – 1. or 2. semester:
Lectures
2
Seminar
1
Tutorial
2
Lab
0
Content (Syllabus outline)

Partial differential equations: Introduction to PDE and examples of partial differential equations of the second order.
Elliptic equations: Poisson's equation. Finite difference method. Discrete maximum principle and global error estimation. Iterative methods for discretized equations. Jacobi, Gauss-Seidel and SOR iterative methods. Symmetric SOR and Chebyshev acceleration. ADI method. Krilov subspace methods. Multigrid methods. Variational methods. Several tipes of finite element methods.
Parabolic equations: Heat transfer equation. Explicit and implicit numerical schemes. Crank-Nicolson's method. Consistency, stability and convergence.
Hyperbolic equations: Wave equation. Characteristics. Characteristical variables. Finite difference method. Courant's condition. Convergence of finite difference approximations for a model equation. Method of characteristics.

Readings

J. Kozak: Numerična analiza, DMFA-založništvo, Ljubljana, 2008.
W. F. Ames: Numerical Methods for Partial Differential Equations, 3rd edition, Academic Press, Boston, 1992.
Z. Bohte: Numerične metode, DMFA-založništvo, Ljubljana, 1991.
S. D. Conte, C. de Boor: Elementary Numerical Analysis : An Algorithmic Approach, 3rd edition, McGraw-Hill, Auckland, 1986.
J. W. Demmel: Uporabna numerična linearna algebra, DMFA-založništvo, Ljubljana, 2000.
E. Isaacson, H. B. Keller: Analysis of Numerical Methods, John Wiley & Sons, New York-London-Sydney, 1966.
D. R. Kincaid, E. W. Cheney: Numerical Analysis : Mathematics of Scientific Computing, 3rd edition, Brooks/Cole, Pacific Grove, 2002.
K. W. Morton, D. F. Mayers: Numerical Solution of Partial Differential Equations, 2nd edition, Cambridge Univ. Press, Cambridge, 2005.
G. D. Smith: Numerical Solution of Partial Differential Equations : Finite Differences Methods, 3rd edition, Clarendon Press, Oxford (New York), 2004.

Objectives and competences

Student supplements knowledge of numerical differentiation, integration and numerical solving of ODE equations. By solving homeworks the obtained theoretical knowledge is consolidated.

Intended learning outcomes

Knowledge and understanding: Understanding of numerical methods for solving partial differential equations. Ability of solving partial differential equations with the computer. Capability of choosing the most appropriate algorithm according to some features of the problem.
Application: Numerical solution of partial differential equations using a computer and error estimation based on theory. Problems that can not be solved any other way that numerically occurs very often
in practise (physics, mechanics, chemistry, economy...).
Reflection: Understanding of theory through applications.
Transferable skills: Skill of using computer for solving numerical problems. Understanding differences between exact and numerical computing. Knowledge of analysis and other fields of mathematics is constructively upgraded.

Learning and teaching methods

Lectures, exercises, homeworks, consultations, project

Assessment

Type (homeworks, examination, oral, coursework, project):

homeworks or project
written exam
oral exam

Grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

Marjetka Krajnc:
JAKLIČ, Gašper, KOZAK, Jernej, KRAJNC, Marjetka, VITRIH, Vito, ŽAGAR, Emil. High order parametric polynomial approximation of conic sections. Constructive approximation, ISSN 0176-4276, 2013, vol. 38, iss. 1, str. 1-18. [COBISS-SI-ID 16716121]
KRAJNC, Marjetka. Interpolation scheme for planar cubic G [sup] 2 spline curves. Acta applicandae mathematicae, ISSN 0167-8019, 2011, vol. 113, no. 2, str. 129-143. [COBISS-SI-ID 16215385]
KRAJNC, Marjetka. Geometric Hermite interpolation by cubic G[sup]1 splines. Nonlinear Analysis, Theory, Methods and Applications, ISSN 0362-546X. [Print ed.], 2009, vol. 70, iss. 7, str. 2614-2626. [COBISS-SI-ID 15508569]
Emil Žagar:
JAKLIČ, Gašper, KOZAK, Jernej, VITRIH, Vito, ŽAGAR, Emil. Lagrange geometric interpolation by rational spatial cubic Bézier curves. Computer Aided Geometric Design, ISSN 0167-8396, 2012, vol. 29, iss. 3-4, str. 175-188. [COBISS-SI-ID 16207449]
KOZAK, Jernej, ŽAGAR, Emil. On geometric interpolation by polynomial curves. SIAM journal on numerical analysis, ISSN 0036-1429, 2004, vol. 42, no. 3, str. 953-967. [COBISS-SI-ID 13398617]
ŽAGAR, Emil. On G [sup] 2 continuous spline interpolation of curves in R [sup] d. BIT, ISSN 0006-3835, 2002, vol. 42, no. 3, str. 670-688. [COBISS-SI-ID 12027993]