We present a new class of numerical methods for solving stochastic differential equations with additive noise on general Riemannian manifolds with high weak order of accuracy. In opposition to the popular approach with projection methods, the proposed methods are intrinsic: they only rely on geometric operations and avoid coordinates and embeddings. We provide a robust and general convergence analysis and an algebraic formalism of exotic planar Butcher series for the computation of order conditions at any high order. To illustrate the methodology, an explicit method of second weak order is introduced, and several numerical experiments confirm the theoretical findings and extend the approach for the sampling of the invariant measure of Riemannian Langevin dynamics.

翻译：本文提出了一类新的数值方法，用于求解一般黎曼流形上具有加性噪声的随机微分方程，该方法具有较高的弱精度阶。与常用的投影方法不同，所提出的方法是内蕴的：它们仅依赖于几何运算，避免了坐标和嵌入的使用。我们提供了鲁棒且通用的收敛性分析，以及用于计算任意高阶阶条件的奇异平面布彻级数的代数形式化描述。为说明该方法，我们引入了一种二阶弱精度的显式方法，并通过若干数值实验验证了理论结果，同时将该方法推广至黎曼朗之万动力学不变测度的采样问题。