  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.
