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

Abstract: In recent years, many positivity-preserving schemes for initial value problems have been constructed by modifying a Runge--Kutta (RK) method by weighting the right-hand side of the system of differential equations with solution-dependent factors. These include the classes of modified Patankar--Runge--Kutta (MPRK) and Geometric Conservative (GeCo) methods. Compared to traditional RK methods, the analysis of accuracy and stability of these methods is more complicated. In this work, we provide a comprehensive and unifying theory of order conditions for such RK-like methods, which differ from original RK schemes in that their coefficients are solution-dependent. The resulting order conditions are themselves solution-dependent and obtained using the 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 of arbitrary order; For MPRK schemes, the order conditions are given implicitly in terms of the stages. From these results, we recover as particular cases all known order conditions from the literature for first- and second-order GeCo as well as first-, second- and third-order MPRK methods. Additionally, we derive sufficient and necessary conditions in an explicit form for 3rd and 4th order GeCo schemes as well as 4th order MPRK methods. We also present a new 4th order MPRK method within this framework and numerically confirm its convergence rate.