Recently a new class of nonlinearly partitioned Runge-Kutta (NPRK) methods was proposed for nonlinearly partitioned systems of ordinary differential equations, $y' = F(y,y)$. The target class of problems are ones in which different scales, stiffnesses, or physics are coupled in a nonlinear way, wherein the desired partition cannot be written in a classical additive or component-wise fashion. Here we use rooted-tree analysis to derive full order conditions for NPRK$_M$ methods, where $M$ denotes the number of nonlinear partitions. Due to the nonlinear coupling and thereby mixed product differentials, it turns out the standard node-colored rooted-tree analysis used in analyzing ODE integrators does not naturally apply. Instead we develop a new edge-colored rooted-tree framework to address the nonlinear coupling. The resulting order conditions are enumerated, provided directly for up to 4th order with $M=2$ and 3rd-order with $M=3$, and related to existing order conditions of additive and partitioned RK methods.

翻译：近期，针对非线性常微分方程组 $y' = F(y,y)$ 提出了一类新型非线性分区龙格-库塔（NPRK）方法。该类问题涉及不同尺度、刚性或物理过程以非线性方式耦合的情形，此时无法通过经典加法或分量方式建立所需分区。本文利用根树分析推导了 NPRK$_M$ 方法的完整阶条件，其中 $M$ 表示非线性分区的数量。由于非线性耦合及其产生的混合乘积微分，用于分析ODE积分器的标准节点着色根树分析不再适用。为此，我们开发了一种新的边着色根树框架来处理非线性耦合。文中列举了所得阶条件，直接给出了 $M=2$ 时最高4阶和 $M=3$ 时最高3阶的条件，并将其与加法及分区RK方法的现有阶条件进行了关联。