In this paper, we develop numerical methods for solving Stochastic Differential Equations (SDEs) with solutions that evolve within a hypercube $D$ in $\mathbb{R}^d$. Our approach is based on a convex combination of two numerical flows, both of which are constructed from positivity preserving methods. The strong convergence of the Euler version of the method is proven to be of order $\tfrac{1}{2}$, and numerical examples are provided to demonstrate that, in some cases, first-order convergence is observed in practice. We compare the Euler and Milstein versions of these new methods to existing domain preservation methods in the literature and observe our methods are robust, more widely applicable and that the error constant is in most cases superior.

翻译：本文针对解在$\mathbb{R}^d$中超立方体$D$内演化的随机微分方程（SDEs）提出了数值求解方法。我们的方法基于两个数值流的凸组合，这两个数值流均采用保持正性的方法构建。证明了该方法欧拉版本的强收敛阶为$\tfrac{1}{2}$，并通过数值算例表明在某些情况下实际观测到一阶收敛。我们将这些新方法的欧拉版本与Milstein版本同文献中现有的域保持方法进行比较，发现我们的方法具有鲁棒性、更广泛的适用性，且在大多数情况下误差常数更优。