  We consider modifications, for example blow ups, of Mori dream spaces and
provide algorithms for investigating the effect on the Cox ring, e.g. testing
finite generation or computing an explicit presentation in terms of generators
and relations. As a first application, we compute the Cox rings of all
Gorenstein log del Pezzo surfaces of Picard number one. Moreover, we show
computationally that all smooth rational surfaces of Picard number at most six
are Mori dream surfaces and we provide explicit presentations of the Cox ring
for those not admitting a torus action. Finally, we provide the Cox rings of
projective spaces blown up at a certain special point configurations.
