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.

翻译：近年来，通过用解依赖因子加权微分方程组右端项来修正龙格-库塔方法，人们构建了许多初值问题的保正格式，包括修正的Patankar-龙格-库塔方法和几何守恒方法。与经典龙格-库塔方法相比，这些方法的精度与稳定性分析更为复杂。本文为这类系数解依赖的龙格-库塔类方法提供了一个全面统一的阶条件理论，其阶条件本身也解依赖，并借助NB-级数理论从标记N-树中直接导出。我们首次给出了任意阶MPRK与GeCo格式的阶条件：对于MPRK格式，阶条件以中间值隐式形式给出。从这些结果中，我们可恢复文献中已知的一阶与二阶GeCo格式、以及一阶、二阶与三阶MPRK方法的全部阶条件作为特例。此外，我们还推导了三阶与四阶GeCo格式以及四阶MPRK方法的显式充分必要条件，并在此框架内提出了一种新的四阶MPRK方法，通过数值实验验证了其收敛阶。