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.

翻译：近年来，通过修改Runge--Kutta (RK)方法，对微分方程组右端项施加解依赖因子加权，构造了许多初值问题的保正格式。这些方法包括改进型Patankar--Runge--Kutta (MPRK) 方法和几何守恒 (GeCo) 方法。与传统RK方法相比，此类方法的精度与稳定性分析更为复杂。本文针对这类系数解依赖的RK类方法（其系数依赖于解，故有别于原始RK格式），提出了一套全面且统一的阶条件理论。由此导出的阶条件本身亦解依赖，并基于NB级数理论获得，因此可通过标记的N-树直接读出。我们首次给出了任意阶MPRK和GeCo格式的阶条件：对于MPRK格式，阶条件以阶段形式隐式给出。基于这些结果，我们作为特例重现了文献中所有已知的一阶和二阶GeCo格式以及一阶、二阶和三阶MPRK方法的阶条件。此外，我们还推导了三阶和四阶GeCo格式以及四阶MPRK方法显式形式的充分必要条件。