A Semi-Lagrangian High-Order Method for Navier-Stokes Equations

We present a semi-Lagrangian method for advection-diffusion and incompressible Navier-Stokes equations. The focus is on constructing stable schemes of second-order temporal accuracy, as this is a crucial element for the successful application of semi-Lagrangian methods to turbulence simulations. We implement the method in the context of unstructured spectral/hp element discretization, which allows for efficient search-interpolation procedures as well as for illumination of the nonmonotonic behavior of the temporal (advection) error of the form: O(Δt^k+{Δx^p+1}/{Δt}). We present numerical results that validate this error estimate for the advection-diffusion equation, and we document that such estimate is also valid for the Navier-Stokes equations at moderate or high Reynolds number. Two- and three-dimensional laminar and transitional flow simulations suggest that semi-Lagrangian schemes are more efficient than their Eulerian counterparts for high-order discretizations on nonuniform grids.