The explicit semi-Lagrangian method for solution of Lagrangian transport equations as developed in Natarajan and Jacobs (2020) is adopted for the solution of stochastic differential equations that is consistent with Discontinuous Spectral Element Method (DSEM) approximations of Eulerian conservation laws. The method extends the favorable properties of DSEM that include its high-order accuracy, its local and boundary fitted properties and its high performance on parallel platforms for the concurrent Monte-Carlo, semi-Lagrangian and Eulerian solution of a class of time-dependent problems that can be described by coupled Eulerian–Lagrangian formulations. Such formulations include the probabilistic models used for the simulation of chemically reacting turbulent flows or particle-laden flows. Consistent with an explicit, DSEM discretization, the semi-Lagrangian method seeds particles at Gauss quadrature collocation nodes within a spectral element. The particles are integrated explicitly in time according to a drift velocity and a Wiener increment forcing and form the nodal basis for an advected interpolant. This interpolant is mapped back in a semi-Lagrangian fashion to the Gauss quadrature points through a least squares fit using constraints for element boundary values. Stochastic Monte-Carlo samples are averaged element-wise on the quadrature nodes. The stable explicit time step Wiener increment is sufficiently small to prevent particles from leaving the element’s bounds. The semi-Lagrangian method is hence local and parallel and does not have the grid complexity, and parallelization challenges of the commonly used Lagrangian particle solvers in particle-mesh methods for solution of Eulerian–Lagrangian formulations. Formal proof is presented that the semi-Lagrangian algorithm evolves the solution according to the Eulerian Fokker–Planck equation. Numerical tests in one and two dimensions for drift–diffusion problems show that the method converges exponentially for constant and non-constant advection and diffusion velocities.

采用 Natarajan 和 Jacobs (2020) 开发的用于求解拉格朗日输运方程的显式半拉格朗日方法来求解随机微分方程，该方法与欧拉守恒定律的不连续谱元方法 (DSEM) 近似一致。该方法扩展了 DSEM 的有利特性，包括其高阶精度、局部和边界拟合特性以及在并行平台上的高性能，用于并发蒙特卡罗、半拉格朗日和欧拉求解一类时间相关问题这可以用耦合的欧拉-拉格朗日公式来描述。此类公式包括用于模拟化学反应湍流或含颗粒流的概率模型。与明确的 DSEM 离散化一致，半拉格朗日方法在谱元素内的高斯正交配置节点处播种粒子。粒子根据漂移速度和维纳增量强迫在时间上明确积分，并形成对流插值的节点基础。该插值以半拉格朗日方式通过最小二乘拟合使用元素边界值的约束映射回高斯正交点。随机蒙特卡罗样本在正交节点上按元素进行平均。稳定显式时间步长维纳增量足够小以防止粒子离开元素的边界。因此半拉格朗日方法是局部并行的，没有网格复杂性，求解欧拉-拉格朗日公式的粒子网格方法中常用拉格朗日粒子求解器的并行化挑战。正式证明半拉格朗日算法根据欧拉福克-普朗克方程演化解。漂移-扩散问题的一维和二维数值测试表明，该方法对于恒定和非常量的对流和扩散速度呈指数收敛。