An explicit semi-Lagrangian, spectral method for solution of Lagrangian transport equations in Eulerian-Lagrangian formulations

Abstract

An explicit high-order semi-Lagrangian method is developed for the solution of Lagrangian transport equations in Eulerian-Lagrangian formulations. The method is consistent with an explicit, discontinuous spectral element method (DSEM) discretization of the Eulerian formulation. The semi-Lagrangian method seeds particles at Gauss quadrature collocation nodes within a spectral element. The particles are integrated explicitly in time 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 and optional constraints for mass and energy preservation. The stable explicit time step of the DSEM solver is sufficiently small to prevent particles seeded at the Gauss quadrature points 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. Numerical tests in one and two dimensions for linear and non-linear advection show that the method converges exponentially. The use of mass and energy constraints can improve accuracy depending on the order of accuracy of the time integrator.