We present a novel numerical method for solving ODEs while preserving polynomial first integrals. The method is based on introducing multiple quadratic auxiliary variables to reformulate the ODE as an equivalent but higher-dimensional ODE with only quadratic integrals to which the midpoint rule is applied. The quadratic auxiliary variables can subsequently be eliminated yielding a midpoint-like method on the original phase space. The resulting method is shown to be a novel discrete gradient method. Furthermore, the averaged vector field method can be obtained as a special case of the proposed method. The method can be extended to higher-order through composition and is illustrated through a number of numerical examples.

翻译：我们提出了一种在保存多元第一组成部分的同时解决数字数的新型方法,其基础是引入多二次辅助变量,将数字数重新表述为等效的、但高度的、只有中点规则适用的四级内分体的等效体。继而消除的二次辅助变量可以在最初的阶段空间上产生中点相似的方法。由此得出的方法被证明是一种新颖的离散梯度方法。此外,平均矢量场方法可以作为拟议方法的一个特例获得。该方法可以通过组成方式扩展至较高顺序,并通过若干数字实例加以说明。