Efficient Newton-Krylov Solver for Aerodynamic Computations

An efficient inexact Newton-Krylov algorithm is presented for the computation of steady two-dimensional aerodynamic flows. The algorithm uses the preconditioned, restarted generalized minimal residual method in matrix-free form to solve the linear system arising at each Newton iteration. The preconditioner is formed using a block-fill incomplete lower-upper factorization of an approximate Jacobian matrix with two levels of fill after applying the reverse Cuthill-McKee reordering. The algorithm has been successfully applied to a wide range of test cases, which include inviscid, laminar, and turbulent aerodynamic flows. In all cases, convergence of the residual to 10 -12 is achieved with a computing cost equivalent to fewer than 1000 function evaluations. The matrix-free inexact Newton-Krylov algorithm is shown to converge faster and more reliably than an approximate Newton algorithm and an approximately factored multigrid algorithm

[1]  W. A. Mulder,et al.  Implicit upwind methods for the Euler equations , 1983 .

[2]  Thomas J. R. Hughes,et al.  A globally convergent matrix-free algorithm for implicit time-marching schemes arising in finite element analysis in fluids , 1991 .

[3]  David E. Keyes,et al.  Application of Newton-Krylov methodology to a three-dimensional unstructured Euler code , 1995 .

[4]  T. Chisholm,et al.  Multigrid acceleration of an approximately-factored algorithm for steady aerodynamic flows , 1997 .

[5]  V. N. Venkatakrishnan,et al.  Implicit Solvers for Unstructured Meshes , 1993 .

[6]  E. Cuthill,et al.  Reducing the bandwidth of sparse symmetric matrices , 1969, ACM '69.

[7]  W. K. Anderson,et al.  Implicit/Multigrid Algorithms for Incompressible Turbulent Flows on Unstructured Grids , 1995 .

[8]  D. Zingg,et al.  Fast Newton-Krylov method for unstructured grids , 1998 .

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

[10]  S Rogers,et al.  A comparison of implicit schemes for the incompressible Navier-Stokes equations with artificial compressibility , 1995 .

[11]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[12]  Yousef Saad,et al.  ILUT: A dual threshold incomplete LU factorization , 1994, Numer. Linear Algebra Appl..

[13]  Dimitri J. Mavriplis,et al.  Multigrid Strategies for Viscous Flow Solvers on Anisotropic Unstructured Meshes , 1997 .

[14]  Youcef Saad,et al.  A Basic Tool Kit for Sparse Matrix Computations , 1990 .

[15]  L. Dutto.  The effect of ordering on preconditioned GMRES algorithm, for solving the compressible Navier-Stokes equations , 1993 .

[16]  Dana A. Knoll,et al.  Comparison of standard and matrix-free implementations of several Newton-Krylov solvers , 1994 .

[17]  A. Pueyo.  An efficient Newton-Krylov method for the Euler and Navier-Stokes equations , 1998 .

[18]  Y. Saad,et al.  Krylov subspace techniques, conjugate gradients, preconditioning and sparse matrix solvers , 1994 .

[19]  Magnus R. Hestenes,et al.  Iterative methods for solving linear equations , 1973 .

[20]  Thomas H. Pulliam,et al.  Recent enhancements to OVERFLOW , 1997 .

[21]  R. Dembo,et al.  INEXACT NEWTON METHODS , 1982 .

[22]  H. Elman.  A stability analysis of incomplete LU factorizations , 1986 .

[23]  Alan George,et al.  The Evolution of the Minimum Degree Ordering Algorithm , 1989, SIAM Rev..

[24]  T. Barth,et al.  An unstructured mesh Newton solver for compressible fluid flow and its parallel implementation , 1995 .

[25]  D. P. Young,et al.  GMRES acceleration of computational fluid dynamics codes , 1985 .

[26]  Homer F. Walker,et al.  Choosing the Forcing Terms in an Inexact Newton Method , 1996, SIAM J. Sci. Comput..