A high-order semi-Lagrangian method for the consistent Monte-Carlo solution of stochastic Lagrangian drift-diffusion models coupled with Eulerian discontinuous spectral element method

TL;DR

This paper introduces a high-order semi-Lagrangian method for solving stochastic Lagrangian drift-diffusion models coupled with Eulerian discontinuous spectral element methods, enabling accurate, parallel, and boundary-fitted solutions.

Contribution

It develops a novel semi-Lagrangian approach that is consistent with DSEM, improving accuracy and parallel performance for coupled Eulerian-Lagrangian stochastic problems.

Findings

Method converges exponentially in 1D and 2D tests.

Ensures particles stay within element bounds with small Wiener increments.

Formal proof links the algorithm to the Eulerian Fokker-Planck equation.

Abstract

The explicit semi-Lagrangian method method for solution of Lagrangian transport equations as developed in [Natarajan and Jacobs, Computer and Fluids, 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. 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.