Convergence Acceleration for Multistage Time-Stepping Schemes

The convergence of a Runge-Kutta (RK) scheme with multigrid is accelerated by preconditioning with a fully implicit operator. With the extended stability of the Runge-Kutta scheme, CFL numbers as high as 1000 could be used. The implicit preconditioner addresses the stiness in the discrete equations associated with stretched meshes. Numerical dissipation operators (based on the Roe scheme, a matrix formulation, and the CUSP scheme) as well as the number of RK stages are considered in evaluating the RK/implicit scheme. Both the numerical and computational eciency of the scheme with the dieren t dissipation operators are discussed. The RK/implicit scheme is used to solve the two-dimensional (2-D) and three-dimensional (3-D) compressible, Reynolds-averaged Navier-Stokes equations. In two dimensions, turbulent o ws over an airfoil at subsonic and transonic conditions are computed. The eects of mesh cell aspect ratio on convergence are investigated for Reynolds numbers between 5:7 10 6 and 100:0 10 6 . Results are also obtained for a transonic wing o w. For both 2-D and 3-D problems, it is demonstrated that the computational time of a well-tuned standard RK scheme can be reduced at least a factor of four.

[1]  Veer N. Vatsa,et al.  Development of an efficient multigrid code for 3-D Navier-Stokes equations , 1989 .

[2]  A. Jameson,et al.  Numerical solution of the Euler equations by finite volume methods using Runge Kutta time stepping schemes , 1981 .

[3]  Antony Jameson,et al.  How Many Steps are Required to Solve the Euler Equations of Steady, Compressible Flow: In Search of a Fast Solution Algorithm , 2001 .

[4]  R. Temam.  Navier-Stokes Equations , 1977 .

[5]  Antony Jameson,et al.  The Evolution of Computational Methods in Aerodynamics , 1983 .

[6]  Rolf Radespiel,et al.  An efficient cell-vertex multigrid scheme for the three-dimensional Navier-Stokes equations , 1989 .

[7]  A. Jameson.  ANALYSIS AND DESIGN OF NUMERICAL SCHEMES FOR GAS DYNAMICS, 1: ARTIFICIAL DIFFUSION, UPWIND BIASING, LIMITERS AND THEIR EFFECT ON ACCURACY AND MULTIGRID CONVERGENCE , 1995 .

[8]  Antony Jameson,et al.  Multigrid algorithms for compressible flow calculations , 1986 .

[9]  R. C. Swanson,et al.  A Study of Multigrid Preconditioners Using Eigensystem Analysis , 2005 .

[10]  Michael B. Giles,et al.  Preconditioned Multigrid Methods for Compressible Flow Calculations on Stretched Meshes , 1997 .

[11]  Eli Turkel,et al.  On Central-Difference and Upwind Schemes , 1992 .

[12]  D. Mavriplis.  Multigrid Strategies for Viscous Flow Solvers on Anisotropic Unstructured Meshes , 1997 .

[13]  Cord-Christian Rossow,et al.  A flux-splitting scheme for compressible and incompressible flows , 2000 .

[14]  W. Mulder.  A new multigrid approach to convection problems , 1989 .

[15]  Hantaek Bae.  Navier-Stokes equations , 1992 .

[16]  Dimitri J. Mavriplis,et al.  On Convergence Acceleration Techniques for Unstructured Meshes , 1998 .

[17]  Jens K. Fassbender,et al.  Improved Robustness for Numerical Simulation of Turbulent Flows around Civil Transport Aircraft at Flight Reynolds Numbers , 2003 .

[18]  J. Blazek.  Verfahren zur Beschleunigung der Lösung der Euler- und Navier-Stokes-Gleichungen bei stationären Über- und Hyperschallströmungen , 1994 .

[19]  R. C. Swanson,et al.  Efficient cell-vertex multigrid scheme for the three-dimensional Navier-Stokes equations , 1990 .

[20]  P. Roe.  Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes , 1997 .

[21]  Wolfgang Hackbusch,et al.  Multi-grid methods and applications , 1985, Springer series in computational mathematics.

[22]  V. Venkatakrishnan,et al.  Uni-directional implicit acceleration techniques for compressible Navier-Stokes solvers , 1999 .

[23]  H. Lomax,et al.  Thin-layer approximation and algebraic model for separated turbulent flows , 1978 .

[24]  Cord-Christian Rossow,et al.  Convergence Acceleration for Solving the Compressible Navier-Stokes Equations , 2006 .

[25]  David Gottlieb,et al.  Optimal time splitting for two- and three-dimensional navier-stokes equations with mixed derivatives , 1981 .

[26]  V. Schmitt,et al.  Pressure distributions on the ONERA M6 wing at transonic Mach numbers , 1979 .

[27]  Bram van Leer,et al.  Design of Optimally Smoothing Multi-Stage Schemes for the Euler Equations , 1989 .

[28]  R. C. Swanson.  On Some Numerical Dissipation Schemes , 1998 .

[29]  R. C. Swanson,et al.  On Central-Difference and Upwind Schemes , 1992 .

[30]  A. Jameson.  Solution of the Euler equations for two dimensional transonic flow by a multigrid method , 1983 .

[31]  R. Swanson,et al.  Multistage Schemes With Multigrid for Euler and Navier-Stokes Equations , 1997 .

[32]  Rolf Radespiel,et al.  Progress with multigrid schemes for hypersonic flow problems , 1995 .

[33]  C. Rossow.  Toward Efficient Computation of Compressible and Incompressible Flows , 2006 .