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(Δtk+Δxp+1Δt). 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.

[1]  J. R. Bates,et al.  Improving the Estimate of the Departure Point Position in a Two-Time Level Semi-Lagrangian and Semi-Implicit Scheme , 1987 .

[2]  B. Sanderson,et al.  Semi‐Lagrangian and COSMIC advection in flows with rotation or deformation , 1999 .

[3]  R. Henderson,et al.  Three-dimensional Floquet stability analysis of the wake of a circular cylinder , 1996, Journal of Fluid Mechanics.

[4]  M. Falcone,et al.  Convergence Analysis for a Class of High-Order Semi-Lagrangian Advection Schemes , 1998 .

[5]  R. W. Davis,et al.  A numerical-experimental study of confined flow around rectangular cylinders , 1984 .

[6]  Janusz A. Pudykiewicz,et al.  Preliminary results From a partial LRTAP model based on an existing meteorological forecast model , 1985 .

[7]  Endre Süli,et al.  Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier-Stokes equations , 1988 .

[8]  Spencer J. Sherwin,et al.  Flow past a square-section cylinder with a wavy stagnation face , 2001, Journal of Fluid Mechanics.

[9]  P. Smolarkiewicz,et al.  A class of semi-Lagrangian approximations for fluids. , 1992 .

[10]  J. P. Huffenus,et al.  A finite element method to solve the Navier‐Stokes equations using the method of characteristics , 1984 .

[11]  J. McGregor,et al.  Economical Determination of Departure Points for Semi-Lagrangian Models , 1993 .

[12]  George Em Karniadakis,et al.  Nodes, modes and flow codes , 1993 .

[13]  Andrei V. Malevsky,et al.  Spline-Characteristic Method for Simulation of Convective Turbulence , 1996 .

[14]  J. M. Watt.  Numerical Initial Value Problems in Ordinary Differential Equations , 1972 .

[15]  U. Ghia,et al.  High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method , 1982 .

[16]  Robert L. Street,et al.  The Lid-Driven Cavity Flow: A Synthesis of Qualitative and Quantitative Observations , 1984 .

[17]  André Robert,et al.  A stable numerical integration scheme for the primitive meteorological equations , 1981 .

[18]  A. Priestley,et al.  Exact projections and the Lagrange-Galerkin method: a realistic alternative to quadrature , 1994 .

[19]  O. Pironneau.  On the transport-diffusion algorithm and its applications to the Navier-Stokes equations , 1982 .

[20]  A. Okajima.  Strouhal numbers of rectangular cylinders , 1982, Journal of Fluid Mechanics.

[21]  A. Staniforth,et al.  Semi-Lagrangian integration schemes for atmospheric models - A review , 1991 .

[22]  Lars Davidson,et al.  Low-Reynolds-number flow around a square cylinder at incidence: study of blockage, onset of vortex shedding and outlet boundary condition , 1998 .

[23]  Stephen J. Thomas,et al.  Parallel algorithms for semi-lagrangian advection , 1997, International Journal for Numerical Methods in Fluids.

[24]  A. Baptista,et al.  A comparison of integration and interpolation Eulerian‐Lagrangian methods , 1995 .

[25]  Rodolfo Bermejo,et al.  Finite element modified method of characteristics for the Navier–Stokes equations , 2000 .

[26]  A. Mcdonald.  Accuracy of Multiply-Upstream, Semi-Lagrangian Advective Schemes , 1984 .

[27]  Jean-Luc Guermond,et al.  Convergence Analysis of a Finite Element Projection/Lagrange-Galerkin Method for the Incompressible Navier-Stokes Equations , 2000, SIAM J. Numer. Anal..

[28]  Francis X. Giraldo,et al.  The Lagrange-Galerkin Spectral Element Method on Unstructured Quadrilateral Grids , 1998 .

[29]  A. Roshko,et al.  Vortex formation in the wake of an oscillating cylinder , 1988 .

[30]  George Em Karniadakis,et al.  A direct numerical simulation of laminar and turbulent flow over riblet-mounted surfaces , 1993, Journal of Fluid Mechanics.

[31]  G. Karniadakis,et al.  Spectral/hp Element Methods for CFD , 1999 .

[32]  Stephen J. Thomas,et al.  The Cost-Effectiveness of Semi-Lagrangian Advection , 1996 .

[33]  B. Schoenung,et al.  NUMERICAL CALCULATION OF LAMINAR VORTEX-SHEDDING FLOW PAST CYLINDERS , 1990 .

[34]  G. Biswas,et al.  TRANSITION AND CHAOS IN TWO-DIMENSIONAL FLOW PAST A SQUARE CYLINDER , 2000 .

[35]  S. Orszag,et al.  High-order splitting methods for the incompressible Navier-Stokes equations , 1991 .

[36]  George Em Karniadakis,et al.  Dynamics and low-dimensionality of a turbulent near wake , 2000, Journal of Fluid Mechanics.

[37]  Jie Shen,et al.  Hopf bifurcation of the unsteady regularized driven cavity flow , 1991 .

[38]  L. Davidson,et al.  Simulation of three-dimensional flow around a square cylinder at moderate Reynolds numbers , 1999 .