Numerical Solution of Ordinary Differential Equations
Code  Completion  Credits  Range 

W01A005  ZK  60B 
 Garant předmětu:
 Luděk Beneš
 Lecturer:
 Luděk Beneš
 Tutor:
 Luděk Beneš
 Supervisor:
 Department of Technical Mathematics
 Synopsis:

Initial value problems for ordinary differential equations and for system of differential equations (onestep and multistep methods). Boundary value problems of secondorder differential equations (finite differences method, shooting method, variational formulation Galerkin) The numerical solution of initialvalue problems in differentialalgebraic equations. Applications.
 Requirements:
 Syllabus of lectures:

1. and 2. lecture:
Revision of M3 = analytical solving of the differential equations. Formulation of the initial value problem for ODE.
The existence and uniqueness of a solution.
3.6. lectures:
The principle of numerical solution IVPI. Onestep methods , methods. Errors, convergence, consistency, absolute stability. The methods of Taylor and RungeKutta type, stabilised RK methods. Using methods, applications. Error estimation, variablestep, RKFehlberg method.
7.  9. lecture:
Linear Multistep Methods (LMM), Errors, convergence, consistency, zero and absolute stability. Numerical methods based on the integration and derivation (AdamsBashforth, AdamsMoulton, BDF), Predictorcorrector methods. Comparison to the onestep methods. Stiff problems. Astability.
10.12. lectures
Boundary value problem of linear and nonlinear second order equations.
The existence and uniqueness of a solution .Shooting methods. Finite difference methods. Existence of a solution of the difference schema, convergence, consistency and stability. Discretization of the boundary conditions. Compact differences.
13. lecture
Variational formulation of the boundary value problems. Generalized and week solutions. Ritz, Galerkin and PetrovGalerkin methods, principle of the finite element method.
14. lecture:
Differentialalgebraic equations (DAE). Basic type of DAE`s. Linear and nonlinear systems. Solvability and the index. Index reduction and constraint stabilization . Onestep methods. Convergence for index 1 and 2 problems.
 Syllabus of tutorials:

The working on the excercises is directly connected to the lectures.
1. and 2. lecture:
Revision of M3 = analytical solving of the differential equations. Formulation of the initial value problem for ODE.
The existence and uniqueness of a solution.
3.6. lectures:
The principle of numerical solution IVPI. Onestep methods , methods. Errors, convergence, consistency, absolute stability. The methods of Taylor and RungeKutta type, stabilised RK methods. Using methods, applications. Error estimation, variablestep, RKFehlberg method.
7.  9. lecture:
Linear Multistep Methods (LMM), Errors, convergence, consistency, zero and absolute stability. Numerical methods based on the integration and derivation (AdamsBashforth, AdamsMoulton, BDF), Predictorcorrector methods. Comparison to the onestep methods. Stiff problems. Astability.
10.12. lectures
Boundary value problem of linear and nonlinear second order equations.
The existence and uniqueness of a solution .Shooting methods. Finite difference methods. Existence of a solution of the difference schema, convergence, consistency and stability. Discretization of the boundary conditions. Compact differences.
13. lecture
Variational formulation of the boundary value problems. Generalized and week solutions. Ritz, Galerkin and PetrovGalerkin methods, principle of the finite element method.
14. lecture:
Differentialalgebraic equations (DAE). Basic type of DAE`s. Linear and nonlinear systems. Solvability and the index. Index reduction and constraint stabilization . Onestep methods. Convergence for index 1 and 2 problems.
 Study Objective:
 Study materials:

R.J. Le Vegue: Finite Difference Methods for Differential Equations.
E. Vitásek: Numerické metody.SNTL 1987
Buchanan J.L.: Numerical Methods and Analysis, McGrawHill, 1992
Nakamura S.: Applied Numerical Methods with Software, Prentice Hall, New York, 1991
Brenan K.E., Campb S.L.,Petzo L.R.: Numerical Solution of InitialValue Problems in DifferentialAlgebraic Equations, Elseview., New York, 1989
Haier E.,Wanner G.:Solving Ordinary Differential Equations II, SpringerVerlag, 1996
 Note:
 Timetable for winter semester 2022/2023:
 Timetable is not available yet
 Timetable for summer semester 2022/2023:
 Timetable is not available yet
 The course is a part of the following study plans: