Numerical Solution of Differential Equations

Course guarantor:

Petr Mayer, Ivana Pultarová

Lecturer:

Petr Mayer

Tutor:

Petr Mayer, Ivana Pultarová

Supervisor:

Department of Mathematics

Synopsis:

After elementary tools of linear algebra (matrix, determinant, Gaussian elimination) are recalled, iterative methods for solving systems of linear algebraic equations are in the focus. Then, the finite difference method and the finite element method are presented and their applications to problems based on differential equations are shown. Attention is also paid to basic methods for solving initial value problems in ordinary differential equations.

Requirements:

Elements of calculus (diffrentiation, integration) are assumed. Elementary knowledge of matrix and vector algebra is appreciated, nevertheless, all necessary tools will be reviewed.

At least 70% excercise class attendance.

Syllabus of lectures:

1. Matrix, inner product of vectors, eigenvalues and eigenvectors of a matrix, spectrum of a matrix.

2. Normed linear space, matrix and vector norms, condition number.

3. Iterative methods for solving systems of linear algebraic equations, sparse matrices.

4. One-step methods for solving initial value problems for ordinary differential equations.

5. Function spaces, inner product of functions, differential operators.

6. Variational principle in 1D problems with positive definite operators, functional of energy, generalized (weak) solution.

7. Approximate solution obtained by variational methods (Ritz method and the finite element method in 1D).

8. Poisson equation in 2D, boundary conditions, applications, Ritz method, finite element method.

9. The finite difference method for 1D boundary value problems and eigenvalue/eigenfuncion problems. Various boundary conditions.

10. Finite difference method for 2D elliptic boundary value problems.

11.Wave equation, numerical solution by the finite difference method, stable and unstable method.

12. Transient heat equation, numerical solution by the finite difference method (in 2D - just for information), stable and unstable method.

13. Reserve

Syllabus of tutorials:

Exercise classes follow the topics of lectures.

Study Objective:

Students will become familiar with basic tools and numerical methods for solving common problems based on differential equations.

Study materials:

!online https://mi21.vsb.cz/sites/mi21.vsb.cz/files/unit/numericke_metody_2.pdf - R. Blaheta: Matematické modelování a metoda konečných prvků, VŠB-TU Ostrava, 2012

!online https://mi21.vsb.cz/sites/mi21.vsb.cz/files/unit/linearni_algebra.pdf - Z. Dostál, V. Vondrák: Lineární algebra, VŠB-TU Ostrava, 2012

!online https://mat.fsv.cvut.cz/chleboun/JCh_vyuka/101MA4/MA4_sbirka21-22S.pdf - J. Chleboun: Příklady k předmětu Matematika 4, FSv ČVUT, Praha, 2021

?H. P. Lantangen, S. Linge: Finite Difference Computing with PDEs, A Modern Software Approach, Springer, Cham, 2017.

?J. C. Butcher: Numerical Methods for Ordinary Differential Equations, John Wiley & Sons, Chichester, 2016.

Note:

Further information:

Viz https://mat.fsv.cvut.cz/vyuka/magistri/zs

Time-table for winter semester 2024/2025:

	06:00–08:0008:00–10:0010:00–12:0012:00–14:0014:00–16:0016:00–18:0018:00–20:0020:00–22:0022:00–24:00
Mon	roomTH:C-206 14:00–15:50 (lecture parallel1) Thákurova 7 (budova FSv) roomTH:B-255 16:00–17:50 (lecture parallel1 parallel nr.101) Thákurova 7 (budova FSv)
Tue
Wed	roomTH:B-255 16:00–17:50 (lecture parallel1 parallel nr.102) Thákurova 7 (budova FSv)
Thu
Fri

Time-table for summer semester 2024/2025:

Time-table is not available yet

The course is a part of the following study plans:

• Stavební Inženýrství - konstrukce a dopravní stavby, specializace Inženýrské konstrukce (compulsory course)
• Stavební Inženýrství - konstrukce a dopravní stavby, specializace Dopravní stavby a geotechnika (compulsory course)