  An explicit high order semi-Lagrangian method is developed for solving
Lagrangian transport equations in Eulerian-Lagrangian formulations. To ensure a
semi-Lagrangian approximation that is consistent with an explicit Eulerian,
discontinuous spectral element method (DSEM) discretization used for the
Eulerian formulation, Lagrangian particles are seeded at Gauss quadrature
collocation nodes within an element. The particles are integrated explicitly in
time to obtain an advected polynomial solution at the advected Gauss quadrature
locations. This approximation 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. An explicit time integration is used for the semi-Lagrangian
approximation that is consistent with the grid based DSEM solver, which ensures
that particles seeded at the Gauss quadrature points do not leave the element's
bounds. The method is hence local and parallel and facilitates the solution of
the Lagrangian formulation without the grid complexity, and parallelization
challenges of a particle solver in particle-mesh methods. Numerical tests with
one and two dimensional advection equation are carried out. The method
converges exponentially. The use of mass and energy constraints can improve
accuracy depending on the accuracy of the time integration.
