 CZECH TECHNICAL UNIVERSITY IN PRAGUE
STUDY PLANS
2023/2024

# Numerical Mathematics

The course is not on the list Without time-table
Code Completion Credits Range Language
2011049 Z,ZK 4 2P+2C Czech
Garant předmětu:
Lecturer:
Tutor:
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 &amp; 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:
Further information:
No time-table has been prepared for this course
The course is a part of the following study plans:
Data valid to 2023-08-30
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