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

TL;DR

This paper develops a boot-strapping technique to create dense output formulae for modified Patankar-Runge-Kutta schemes, maintaining conservation and positivity, and demonstrates higher order accuracy with reduced computational effort.

Contribution

It introduces a novel boot-strapping approach to design dense output formulae for MPRK schemes, enabling higher order accuracy while preserving key properties.

Findings

Constructed explicit and implicit dense output formulae up to order three.

Demonstrated higher order dense output formulae maintain conservation and positivity.

Showed that the approach reduces computational effort despite solving linear systems.

Abstract

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.