We propose a novel way to describe numerical methods for ordinary differential equations 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. Moreover, these multi-indices B-series characterise uniquely the Taylor development of local and affine equivariant maps.

翻译：我们提出了一种通过多指标概念描述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级数中的有根树。多指标最近被引入用于描述奇异随机偏微分方程的解，这种从有根树的组合学转变使得数值格式的描述更为简洁。此外，此类多指标B级数唯一地表征了局部仿射等变映射的泰勒展开。