Positivity-preserving schemes gained more and more attention in recent years. However, higher order numerical schemes that unconditionally preserve the positivity do not belong to the class of general linear methods, i.e. the resulting iteration scheme is nonlinear even when applied to a linear system of differential equations. This fact has far-reaching consequences and results in complicating their analysis with regard to order of accuracy and stability. Also, equipping positivity-preserving methods with dense output formulae, adaptive time-step control or a relaxation procedure to ensure entropy stability rise additional challenges. For instance, the numerical approximations computed by a dense output formula or a relaxation method should also be positivity-preserving.

In this talk, we intoduce and explore one class of unconditionally positivity-preserving schemes, the so-called modified Patankar-Runge-Kutta (MPRK) schemes, and point out their major benefits as well as challenges and ideas for their analysis. To this end, we provide insights into current and an outlook on future research on how to equip MPRK methods with the above algorithms. We perform numerical experiments on all these topics to validate and test our findings.