Numerical Mathematics

Course guarantor:

Petr Sváček

Lecturer:

Luděk Beneš, Tomáš Bodnár, Marta Čertíková, Jiří Fürst, Lukáš Hájek, Jiří Holman, Vladimír Hric, Jan Karel, Radka Keslerová, Matěj Klíma, Petr Louda, Olga Majlingová, Vladimír Prokop, Hynek Řezníček, Petr Sváček, David Trdlička, Jan Valášek

Tutor:

Luděk Beneš, Tomáš Bodnár, Marta Čertíková, Jiří Fürst, Lukáš Hájek, Tomáš Halada, Tomáš Hlavatý, Jiří Holman, Vladimír Hric, Jan Karel, Radka Keslerová, Matěj Klíma, Patrik Kovář, Ondřej Krejčí, Anna Lancmanová, Petr Louda, Pavel Mačák, Olga Majlingová, Josef Musil, Tomáš Neustupa, Vladimír Prokop, Prokop Pučejdl, Vítězslav Putna, Hynek Řezníček, Petr Sváček, Adam Tater, David Trdlička, Karel Vacek, Jan Valášek

Supervisor:

Department of Technical Mathematics

Synopsis:

Numerical solution of systems of linear equations, iterative methods. Numerical solution of nonlinear algebraic equations.

Least squares method. Numerical solution of ordinary differential equations,

initial and boundary value problems. Numerical solution of basic linear

partial differential equations by finite difference method.

Requirements:

Syllabus of lectures:

1. Norm and spectral radius of matrices. Principle of iterative methods. Fixed

point iterative method.

2. Jacobi and Gauss-Seidel iteration method. Convergence.

3. Minimization of function and gradient methods. Steepest descent

method. Least squares method and system of normal equations.

4. Systems of nonlinear equations, existence and uniqueness of

solutions. Contract mapping and fixed point iterations. Newton's method.

5. Numerical solution of ordinary differential equations (ODE). Explicit and implicit Euler

method. Collatz method.

6. One-step methods, local discretization error, global error, order of the method.

7. One-step methods of Runge-Kutty type. Higher order methods.

8. Boundary value problem for 2nd order linear ODEs in selfadjoint form. Existence a

solution uniqueness. Numerical solution of the problem by the finite

difference method.

9. Principle of 2D finite difference methods. Taylor expansion. Boundary value problem for

Poisson equation with Dirichlet condition.

10.-11. Initial boundary value problem for heat conduction equation. Numerical

solution of heat conduction problem, explicit and implicit scheme. Convergence and stability of the method.

12.-13. Initial boundary value problem problem for wave equation. Numerical

solution by finite difference method, explicit and

implicit scheme. Convergence and stability of the method.

Syllabus of tutorials:

1. Norm and spectral radius of matrices. Principle of iterative methods. Fixed

point iterative method.

2. Jacobi and Gauss-Seidel iteration method. Convergence.

3. Minimization of function and gradient methods. Steepest descent

method. Least squares method and system of normal equations.

4. Systems of nonlinear equations, existence and uniqueness of

solutions. Contract mapping and fixed point iterations. Newton's method.

5. Numerical solution of ordinary differential equations (ODE). Explicit and implicit Euler

method. Collatz method.

6. One-step methods, local discretization error, global error, order of the method.

7. One-step methods of Runge-Kutty type. Higher order methods.

8. Boundary value problem for 2nd order linear ODEs in selfadjoint form. Existence a

solution uniqueness. Numerical solution of the problem by the finite

difference method.

9. Principle of 2D finite difference methods. Taylor expansion. Boundary value problem for

Poisson equation with Dirichlet condition.

10.-11. Initial boundary value problem for heat conduction equation. Numerical

solution of heat conduction problem, explicit and implicit scheme. Convergence and stability of the method.

12.-13. Initial boundary value problem problem for wave equation. Numerical

solution by finite difference method, explicit and

implicit scheme. Convergence and stability of the method.

Study Objective:

1. Matrices; System of linear equations - direct methods; Gauss elimination for tri-diagonal systems; Principle of iterative methods; norms and spectral radius.,

2. Simple and Jacobi iterative method; Gauss-Seidel method; convergence conditions.,

3. Systems of nonlinear equations; Problems of existence and uniqueness of the solution; Iterative methods - Newton method; Analogy of 1D problem.,

4. Principle of interpolation; Interpolation by algebraic polynomials; Existence and uniqueness of the polynomial; Interpolation by spline functions; Advantages of this interpolation; Practical applications.,

5. Least squares approximation - principle of approximation by an algebraic polynomial; Derivation of the system of normal equations;,

6-8. Numerical solution of the Cauchy problem for the 1st order equation and for a system in normal form; Cauchy problem for the nth order equation; Principle of one-step methods of Euler & Runge-Kutta; Convergence; Practical application;,

9-10. The problems of the solution of the boundary value problems for an 2nd order ordinary differential equation, comparison with the Cauchy problem; Existence and uniqueness; Dirichlet problem; Principle of the mesh methods (finite difference methods), convergence; Existence and uniqueness of the solution of the associated system of linear equations; Shooting method;,

11-13. Numerical solution of the linear partial differential 2nd order equations in 2D -mesh methods; Classes of equations; Formulation of elementary problems for the equations of the mathematical physics (Laplace and Poisson equation; Heat transfer equation, Wave equation); Difference substitutions of the first and second derivative order of the approximation; Principle of the mesh method for the solution of individual types of problems; Convergence and stability;

Study materials:

1. Mathews, J. H.: Numerical Methods for Mathematics, Science and Engineering, Prentice Hall International, 2nd edition,1992, 2. Gerald, C.F., Wheatley, P.O.: Applied Numerical Analysis, Addison Wesley, 6th edition, 1999

Note:

Time-table for winter semester 2024/2025:

Time-table is not available yet

Time-table for summer 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	roomT4:D1-366 10:45–12:15 (lecture parallel1) Dejvice
Tue
Wed
Thu	roomT4:A1-405a 07:15–08:45 (parallel nr.1) Dejvice roomT4:A1-405a 09:00–10:30 (parallel nr.2) Dejvice roomT4:A1-405a 10:45–12:15 (parallel nr.3) Dejvice roomT4:A1-405a 12:30–14:00 (parallel nr.4) Dejvice roomT4:A1-405a 14:15–15:45 (parallel nr.5) Dejvice roomT4:A1-405a 16:00–17:30 (parallel nr.6) Dejvice roomT4:A1-405a 17:45–19:15 (parallel nr.7) Dejvice
Fri	roomT4:A1-405a 07:15–08:45 (parallel nr.8) Dejvice roomT4:A1-405a 09:00–10:30 (parallel nr.9) Dejvice roomT4:A1-405a 10:45–12:15 (parallel nr.10) Dejvice

The course is a part of the following study plans:

• 10 62 67 00 BTZI 2012 P základ (compulsory course in the program)
• 11 68 73 00 BTZI 2012 K základ (compulsory course in the program)
• 02 26 31 34 BSTR EPT 2012 P základ (compulsory course in the program)
• 03 26 31 36 BSTR IAT 2012 P základ (compulsory course in the program)
• 04 26 31 38 BSTR KPP 2012 P základ (compulsory course in the program)
• 06 40 45 48 BSTR EPT 2012 K základ (compulsory course in the program)
• 07 40 45 50 BSTR IAT 2012 K základ (compulsory course in the program)
• 08 40 45 52 BSTR KPP 2012 K základ (compulsory course in the program)
• 05 40 45 46 BSTR TZP 2012 K základ (compulsory course in the program)
• 05 40 45 46 DSTR TZP 2012 K základ (compulsory course in the program)
• 06 40 45 48 DSTR EPT 2012 K základ (compulsory course in the program)
• 07 40 45 50 DSTR IAT 2012 K základ (compulsory course in the program)
• 08 40 45 52 DSTR KPP 2012 K základ (compulsory course in the program)
• 10 62 67 00 DTZI 2012 P základ (compulsory course in the program)
• 11 68 73 00 DTZI 2012 K základ (compulsory course in the program)