We consider a generic and explicit tamed Euler--Maruyama scheme for multidimensional time-inhomogeneous stochastic differential equations with multiplicative Brownian noise. The diffusion coefficient is uniformly elliptic, H\"older continuous and weakly differentiable in the spatial variables while the drift satisfies the Ladyzhenskaya--Prodi--Serrin condition, as considered by Krylov and R\"ockner (2005). In the discrete scheme, the drift is tamed by replacing it by an approximation. A strong rate of convergence of the scheme is provided in terms of the approximation error of the drift in a suitable and possibly very weak topology. A few examples of approximating drifts are discussed in detail. The parameters of the approximating drifts can vary and be fine-tuned to achieve the standard 1/2-strong convergence rate with a logarithmic factor. The result is then applied to provide numerical solutions for stochastic transport equations with singular vector fields satisfying the aforementioned condition.
翻译:我们考虑了一种通用的、明确的有色Euler-Maruyama方案,用于使用多种复制性的棕色噪音的多时间性随机差异方程式。扩散系数在空间变量中是均匀的椭圆、H\"老的连续和微小的差别,而漂移满足了Ladyzenskaya-Prodi-Serrin条件,Krylov和R\'ockner (2005年)认为,漂移满足了Ladyzenskaya-Prodi-Serrin条件。在离散的方案中,通过近似取代它来调节漂移。该方法的高度趋同率以适合且可能非常弱的表层漂移的近似误差来表示。详细讨论了近似漂移的几个例子。相近的漂移参数可以变化,并经过微调,以达到标准1/2强的趋同率和对数系数。然后将结果用于为与符合上述条件的单一矢量场的单向式传输方程式提供数字解决办法。