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In recent years, many positivity-preserving schemes for initial valueproblems have been constructed by modifying a Runge--Kutta (RK) method byweighting the right-hand side of the system of differential equations withsolution-dependent factors. These include the classes of modifiedPatankar--Runge--Kutta (MPRK) and Geometric Conservative (GeCo) methods.Compared to traditional RK methods, the analysis of accuracy and stability ofthese methods is more complicated. In this work, we provide a comprehensive andunifying theory of order conditions for such RK-like methods, which differ fromoriginal RK schemes in that their coefficients are solution-dependent. Theresulting order conditions are themselves solution-dependent and obtained usingthe theory of NB-series, and thus, can easily be read off from labeled N-trees.We present for the first time order conditions for MPRK and GeCo schemes ofarbitrary order; For MPRK schemes, the order conditions are given implicitly interms of the stages. From these results, we recover as particular cases allknown order conditions from the literature for first- and second-order GeCo aswell as first-, second- and third-order MPRK methods. Additionally, we derivesufficient and necessary conditions in an explicit form for 3rd and 4th orderGeCo schemes as well as 4th order MPRK methods.

Order conditions for Runge--Kutta-like methods with solution-dependent coefficients