Stochastic differential equations projected onto manifolds occur widely in physics, chemistry, biology, engineering, nanotechnology and optimization theory. In some problems one can use an intrinsic coordinate system on the manifold, but this is often computationally impractical. Numerical projections are preferable in many cases. We derive an algorithm to solve these, using adiabatic elimination and a constraining potential. We also review earlier proposed algorithms. Our hybrid midpoint projection algorithm uses a midpoint projection on a tangent manifold, combined with a normal projection to satisfy the constraints. We show from numerical examples on spheroidal and hyperboloidal surfaces that this has greatly reduced errors compared to earlier methods using either a hybrid Euler with tangential and normal projections or purely tangential derivative methods. Our technique can handle multiple constraints. This allows, for example, the treatment of manifolds that embody several conserved quantities. The resulting algorithm is accurate, relatively simple to implement and efficient.

翻译：在物理、化学、生物学、工程、纳米技术和优化理论中,预测到元体的电磁微分方程式会广泛出现。在有些问题上,人们可以使用对元体的内在协调系统,但这往往在计算上是不切实际的。在许多情况下,数字预测是可取的。我们利用非对称消除和抑制潜力来得出解决这些问题的算法。我们还审查了早先提出的算法。我们的混合中点预测法对正切的元件进行中点预测,同时进行正常的预测,以满足这些限制。我们从关于近亲和超双球表面的数字例子中可以看出,与早先使用混合的电动器和正正态预测或纯正值衍生法的方法相比,这大大减少了错误。我们的技术可以处理多种限制因素,例如,可以处理体现若干节量的元件。由此产生的算法准确、相对简单、执行和高效。