Differential Equations on Computer
Code  Completion  Credits  Range  Language 

12DRP  Z,ZK  5  2+2  Czech 
 Garant předmětu:
 Richard Liska
 Lecturer:
 Richard Liska
 Tutor:
 Richard Liska
 Supervisor:
 Department of Laser Physics and Photonics
 Synopsis:

Ordinary differential equations, analytical methods; Ordinary differential equations, numerical methods, RungeKutta methods, stability; Partial differential equations, analysis, hyperbolik, parabolic and elliptic equations, posedness of differential equaitons; Partial differential equations, numerical solution, finite difference
methods, difference schemes, order of approximation, stability, convergence, modified equation, diffusion, dispersion; Conservation laws and their numerical solution, shallow water equations, Euler equations, Lagrangian methods, ALE methods; Practical computation in
Matlab system for numerics and Maple for analysis of schemes.
 Requirements:
 Syllabus of lectures:

1. Ordinary differential equations, analytical methods, stability.
2. Ordinary differential equations, RungeKutta methods, stability function, stability domain, order of method.
3. Ordinary differential equations with boundary conditions.
4. Hyperbolic partial differential equations, characteristics, boundary conditions, finite difference methods
5. Convergence, consistency, wellposedness, stability, LaxRichtmyer theorem, CourantFriedrichsLewy (CFL) condition.
6. Fourier analysis of wellposedness and stability, von Neumann stability condition.
7. LaxWendroff scheme, implicit schemes, order of approximation, modified equation, diffusion, dispersion.
8. Parabolic equations, difference schemes for parabolic equations.
9. Elliptic equations, iterative methods for solving systems of linear equations.
10. Advection equation in 2D, dimensional spliting, difference schemes.
11. Conservation laws, integral form, RankinHugoniot condition.
12. Burgers equation, shallow water equations, Euler equations, shock wave. rarefaction wave, contact discontinuity, difference schemes.
13. Lagrangian methods for Euler equations, mass coordinates.
14. ALE (Arbitrary LagrangianEulerian) method, mesh smoothing, remapping.
 Syllabus of tutorials:

1. Ordinary differential equations, analytical methods, stability.
2. Ordinary differential equations, design of RungeKutta (RK) methods.
3. Computing stability function and domain of RK method, order of RK method.
4. Finite difference schemes for advection equation, numerical verification of their properties  stability and order of approximation.
5. Analytical determination of order of approximation of difference scheme.
6. Analytical determination of stability condition by Fourier method.
7. Analyticalnumerical determination of stability condition by Fourier method.
8. Computing modified equation of difference scheme.
9. Difference schemes for parabolic equation  heat equation.
10. Difference schemes for advection diffusion equation.
11. Difference schemes for elliptic Poisson equation.
12. Test  design and analysis of finite difference scheme.
13. Difference schemes for Burgers equation, shallow water equation and Euler equations.
14. Lagrangian difference schemes, ALE method.
 Study Objective:

Knowledge:
Knowledge of numerical solution of differential equations.
Skills:
Ability to design and analyze numerical methods for solution of differential equations.
 Study materials:

Key references:
[1] J.C. Strikwerda: Finite Difference Schemes and Partial Differential Equations, Chapman and Hall, New York, 1989.
Recommended references:
[2] R.J. LeVeque: Numerical Methods for Conservation Laws, Birkhauser Verlag, Basel, 1990.
Study aids:
Computer classroom Unix with integrated mathematical systems Matlab and Maple.
 Note:
 Timetable for winter semester 2023/2024:
 Timetable is not available yet
 Timetable for summer semester 2023/2024:
 Timetable is not available yet
 The course is a part of the following study plans:

 Matematické inženýrství (elective course)
 Fyzika a technika termojaderné fúze (elective course)
 Aplikovaná algebra a analýza (elective course)
 Fyzika plazmatu a termojaderné fúze (elective course)
 Matematické inženýrství (elective course)
 Fyzikální elektronika  Počítačová fyzika (PS)