Dr. Mario Helm TU Bergakademie Freiberg

Dr. Mario Helm TU Bergakademie Freiberg

Steffen Pacholak Summer Semester 2015

Numerical Analysis of Differential Equations

Questions for selftest and repetition

1. How can a higher order ordinary differential equation or a system of such equations be reduced to a system of first order ODEs? Why is this transformation that important for the numerical solution of higher order problems? Illustrate the transformation by an example of your choice.

2. What are the method’s equations for the explicit and the implicit Euler method? Illustrate the work with both methods on an example of your choice, e. g. by calculation of the first two approximations in each case.

3. How can a given Butcher tableau be translated into the set of formulae of the associated RKM? Illustrate this on an example of your choice.

4. What are the advantages and disadvantages of explicit and implicit methods. Explain the relation to the answers of the questions 13 and 14.

5. What do you understand by the local and the global discretization error and the order of consistency of a method?

6. How can the order of consistency of an explizit RKM be checked (maximal 3-stage is suffi- cient)?

7. Assume that the (constant and sufficiently small) step size h of a RKM with order 4 is reduced by a factor 10. Which behaviour of the error of the numerical approximation would you expect?

8. How does step size control work (idea is sufficient)? In which methods, a step size control can be implemented particularly efficient?

9. What do you understand by a linear multistep method? Give one or two examples.

10. What are the advantages and disadvantages of LMSM in comparison to OSM/RKM?

11. What do you understand by zero stability of a LMSM? How are consistency, zero stability and convergence connected? How can the consistency of a LMSM be checked?

12. Explain the concept of absolute stability. Which role do Dahlquist’s test equation and the stability function play in this context?

13. The stability function of an explicit m-stage RKM is given by

R(ˆ h) =

p

X

j=0

1 j! ˆ h

+

m

X

j=p+1

ˆ h

b

A

e,

where A and b are token from the Butcher-Tableau and e is a vector with all entries one.

Why can such an explicit method never be absolutely stable?

14. Describe the phenomenon of „stiffness “ of an ODE. Discuss whether/when explicit or implicit methods should be used for the numerical solution of a stiff ODE.

15. How can linear PDEs with symmetric coefficient matrix be classified? Give examples for the three most important classes.

16. What is understood by a well posed problem in the sense of Hadamard?

17. Which methods are useful to construct test problems with analytic solution for elliptic BVPs/

parabolic IBVPs?

18. Explain how a 2D Dirichlet problem for the Laplace/Poisson equation can be discretized with finite differences. You can restrict yourself to a simple domain (square/rectangle) and a uniform grid. What is the order of consistency for this discretization?

19. How can the discretization of the 2D Laplacian on the unit square be constructed from the corresponding 1D version?

20. Describe two variants for the discretization of Neumann boundary conditions (problem corre- sponding to question 18). What are the advantages/disadvantages for each variant? Illustrate both variants on an 1D example of your choice.

21. Discretize the IBVP for the 1D heat equation (infinitely thin rod with length one) in space direction. Formulate the arising system of linear ODEs as well as the associated initial con- dition.

22. Explain how the IVP from question 21 can be solved numerically. Consider all three schemes, that were treated during the lecture.

23. What are the consistency orders for the three schemes from question 22? Which refinement

curves make sense in this context? Which additional assumption has to be fulfilled by the

refinement curve in the explicit Euler scheme?