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A semi-Lagrangian high-order method for Navier-Stokes equations. (English) Zbl 1028.76026
Summary: 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 $𝒪\left({\Delta }{t}^{k}+\frac{{\Delta }{x}^{p+1}}{{\Delta }t}\right)$. 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.
##### MSC:
 76M10 Finite element methods (fluid mechanics) 76M22 Spectral methods (fluid mechanics) 76D05 Navier-Stokes equations (fluid dynamics)