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Zbl 1028.76026
Xiu, Dongbin; Karniadakis, George Em
A semi-Lagrangian high-order method for Navier-Stokes equations. (English)
[J] J. Comput. Phys. 172, No.2, 658-684 (2001). ISSN 0021-9991

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 ${\cal O}(\Delta t^k+{ \Delta x^{p+1} \over\Delta t})$. 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 2000:: *76M10 Finite element methods
76M22 Spectral methods
76D05 Navier-Stokes equations (fluid dynamics)

Keywords: semi-Lagrangian spectral/hp element method; three-dimensional incompressible Navier-Stokes equations; second-order temporal accuracy; search-interpolation procedures; advection-diffusion equation

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