We propose a novel way to study numerical methods for ordinary differential equations in one dimension via the notion of multi-indice. The main idea is to replace rooted trees in Butcher's B-series by multi-indices. The latter were introduced recently in the context of describing solutions of singular stochastic partial differential equations. The combinatorial shift away from rooted trees allows for a compressed description of numerical schemes. Furthermore, such multi-indices B-series uniquely characterize the Taylor expansion of one-dimensional local and affine equivariant maps.

翻译：我们提出了一种通过多指标概念研究一维常微分方程数值方法的新途径。核心思想是用多指标替代Butcher B级数中的根树结构。此类多指标是近期在描述奇异随机偏微分方程解的背景下提出的。从根树到多指标的组合转换实现了数值格式的压缩描述。此外，这类多指标B级数能唯一表征一维局部且仿射等变映射的泰勒展开式。