Differential Equations on Computer

Garant předmětu:

Richard Liska

Lecturer:

Richard Liska

Tutor:

Richard Liska

Supervisor:

Department of Physical Electronics

Synopsis:

Ordinary differential equations, analytical methods; Ordinary differential equations, numerical methods, Runge-Kutta 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, Runge-Kutta 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, well-posedness, stability, Lax-Richtmyer theorem, Courant-Friedrichs-Lewy (CFL) condition.

6. Fourier analysis of well-posedness and stability, von Neumann stability condition.

7. Lax-Wendroff 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, Rankin-Hugoniot 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 Lagrangian-Eulerian) method, mesh smoothing, remapping.

Syllabus of tutorials:

1. Ordinary differential equations, analytical methods, stability.

2. Ordinary differential equations, design of Runge-Kutta (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. Analytical-numerical 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.

http://www-troja.fjfi.cvut.cz/~liska/drp

Note:

Time-table for winter semester 2023/2024:

Time-table is not available yet

Time-table for summer semester 2023/2024:

Time-table 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)