A Boot-Strapping Technique to Design Dense Output Formulae for Modified Patankar-Runge-Kutta Methods

In this work modified Patankar-Runge-Kutta (MPRK) schemes up to order four are considered and equipped with a dense output formula of appropriate accuracy. Since these time integrators are conservative and positivity preserving for any time step size, we impose the same requirements on the corresponding dense output formula. In particular, we discover that there is an explicit first order formula. However, to develop a boot-strapping technique we propose to use implicit formulae which naturally fit into the framework of MPRK schemes. In particular, if lower order MPRK schemes are used to construct methods of higher order, the same can be donw with the dense output formulae we propose in this work. We explicitly construct formulae up to order three and demonstrate how to generalize this approach as long as the underlying Runge-Kutta method possesses a dense output formulae of appropriate accuracy. We also note that even though linear systems have to be solved to compute an approximation for intermediate points in time using these higher order dense output formulae, the overall computational effort is reduced compared to using the scheme with a smaller step size.