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Modified Patankar-Runge-Kutta (MPRK) schemes are numerical methods for thesolution of positive and conservative production-destruction systems. Theyadapt explicit Runge-Kutta schemes to ensure positivity and conservationirrespective of the time step size.  The first two members of this class were introduced in [Burchard et. al.: Ahigh-order conservative Patankar-type discretisation for stiff systems ofproduction-destruction equations. Appl. Numer. Math., 47(1):1-30, 2003] andhave been successfully applied in a large number of applications. Recently, weintroduced a general definition of MPRK schemes and presented a thoroughinvestigation of first and second order MPRK schemes in [Kopecz. S, Meister,A.: On order conditions for modified patankar-runge-kutta schemes.arXiv:1702.04589 [math.NA], 2017.].  A potentially third order Patankar-type method was introduced in [FormaggiaL., Scotti, A.: Positivity and conservation properties of some integrationschemes for mass action kinetics. SIAM J. Numer. Anal., 49(3):1267-1288,2011.]. This method uses the MPRK22(1) scheme of [Burchard et. al.: Ahigh-order conservative Patankar-type discretisation for stiff systems ofproduction-destruction equations. Appl. Numer. Math., 47(1):1-30, 2003] as apredictor and a modification of the BDF(3) multistep method as a corrector.This method is at most third order and at least second order accurate.  In this paper we continue the work of [Kopecz. S, Meister, A.: On orderconditions for modified patankar-runge-kutta schemes. arXiv:1702.04589[math.NA], 2017.] and present necessary and sufficient conditions for thirdorder MPRK schemes. For the first time, we introduce MPRK schemes, which arethird order accurate independent of the specific positive and conservativesystem under consideration. The theoretical results are confirmed by numericalexperiments.

Unconditionally Positive and Conservative Third Order Modified Patankar-Runge-Kutta Discretizations of Production-Destruction Systems