Stability of approximate factorization with $ heta $-methods

Approximate factorization seems for certain problems a viable alternative to time splitting. Since a splitting error is avoided, accuracy will in general be favourable compared to time splitting methods. However, it is not clear to what extent stability is affected by factorization. Therefore we study here the effects of factorization on a simple, low order method, namely the $theta$-method. For this simple method it is possible to obtain rather precise results, showing limitations of the approximate factorization approach.