ELUSAKIN, O. BDF a BBDF pro řešení diferenciálních rovnic [online]. Brno: Vysoké učení technické v Brně. Fakulta strojního inženýrství. 2024.

Posudek vedoucího

Zatočilová, Jitka

The present thesis is devoted to the backward differentiation formulas (BDF) and block backward differentiation formulas (BBDF), which are used for solving stiff differential equations. I think the student could have spent more time on the thesis, especially since she was allowed to submit the thesis 4 months later than the regular deadline. Thus, the thesis contains unnecessary errors (e.g. on p. 7 missing parentheses in P'_3(t), in Lemma 2.3.1 there is a reference to the relation (3.3.10) instead of (2.3.10), on p. 13 is an incorrect reference to the literature, on p. 19 the signs in matrix B are wrong, on p. 33 the 0-stability conditions are wrong, on p. 37 rho is wrong, in test problem 4 is misspelt initial condition,... ) and shortcomings among which I would mention the lack of distinction between 0-stability and A-stability in BBDF methods, where in addition the A-stability region is not plotted in most methods, then the essentially missing commentary on the numerical results, where moreover, in choosing a relatively large time step h, the contribution of BBDF methods is not very obvious..., and the bibliography also has significant shortcomings. The contribution of this thesis is the derivation of the error constants and stability polynomials for some BBDF methods and the programming of all these methods in Matlab. Despite these criticisms, I conclude that the objectives were fulfilled and recommend the thesis for defence.

Posudek oponenta

Tomášek, Petr

The thesis focuses on backward differentiation formula (BDF) methods and block backward differentiation formula (BBDF) methods for initial value problems of ordinary differential equations (ODE). Several methods are derived for the both approaches. There are mentioned notions of method accuracy, A-stability and D-stability. There are derived accuracy orders for the introduced methods in the thesis. There is quite a lot of grammatical mistakes in the thesis. Moreover, the errors appear in mathematical formulae and notions, e.g. p.7: brackets are missing in $P'_3(t)$, which turns it incorrect. p. 11: bad references in Lemma 2.3.1 and Figure 2.1. p. 12. Definition of A-stability does not correspond to the well-known A-stability concept. p. 33. second line - relation for $|r_j|$ There are also deficiencies in references, namely incomplete bibliographic data in most of them, even missing authors in [16]. The goals of the thesis are partially met. The results would be much more significant if the number of steps is chosen much greater (e.g. 4000). Moreover, the test problem 4 has wrong accuracy results due to incorrect exact solution (in the thesis and also in matlab codes). The results mentioned in the text are incorrect also with respect to another stepsize h used. It is due to wrong parameter in linspace command, where the third parameter is number of nodes in vector, not number of steps. Hence the actual stepsize is a bit greater than the introduced ones. I have not found computer specifications on which the tests were made (hardware, operational system) The elapsed times introduced in the thesis are then less relevant. On the other hand, student has implemented the introduced methods in matlab and demonstrated their usage for ODE IVPs. Finally, I recommend the thesis for a defense with overall evaluation D - satisfactory.

eVSKP id 162483