Implicit upwind residual-distribution Euler and Navier-Stokes solver on unstructured meshes

Implicit iterative solution techniques are considered for application to a compressible Euler and Navier-Stokes solver using upwind residual-distribution schemes on unstructured meshes. Numerical evaluation of the complete Jacobian matrix needed for the linearization process is achieved at low cost, either by finite difference approximation or by Broyden's update. It enables nonlinear solution strategies such as Newton iterative methods where linear systems are solved approximately using an accelerated iterative scheme. The linearized backward Euler scheme is used to integrate the discretized equations in time, together with a simple time-step evolution strategy. Alternatively, when this strategy fails, it is possible to use a fixed-point acceleration method that has proven quite robust. Numerical applications show the efficiency of the iterative strategy for various flow conditions.

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

[2]  N. Ron-Ho,et al.  A Multiple-Grid Scheme for Solving the Euler Equations , 1982 .

[3]  John E. Dennis,et al.  Numerical methods for unconstrained optimization and nonlinear equations , 1983, Prentice Hall series in computational mathematics.

[4]  P. Roe.  The 'optimum' upwind advection on a triangular mesh , 1990 .

[5]  C. Lanczos.  Solution of Systems of Linear Equations by Minimized Iterations1 , 1952 .

[6]  V. Venkatakrishnan.  Preconditioned conjugate gradient methods for the compressible Navier-Stokes equations , 1990 .

[7]  R. V. Buuren,et al.  Instabilities of Stationary Inviscid Compressible Flow around an Airfoil , 1997 .

[8]  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 .

[9]  R. Freund,et al.  QMR: a quasi-minimal residual method for non-Hermitian linear systems , 1991 .

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

[11]  Gérard Degrez,et al.  Implicit iterative methods for a multidimensional upwind Euler /Navier-Stokes solver on unstructured meshes , 1995 .

[12]  P. Sonneveld.  CGS, A Fast Lanczos-Type Solver for Nonsymmetric Linear systems , 1989 .

[13]  Philip L. Roe,et al.  Conservative upwind residual-distribution schemes based on the steady characteristics of the euler equations , 1995 .

[14]  Yousef Saad,et al.  Hybrid Krylov Methods for Nonlinear Systems of Equations , 1990, SIAM J. Sci. Comput..

[15]  P. Orkwis,et al.  Newton's Method Solver for the Axisymmetric Navier-Stokes Equations , 1992 .

[16]  Dimitri J. Mavriplis,et al.  Implicit Solvers for Unstructured Meshes , 1993 .

[17]  Philip L. Roe,et al.  A multidimensional generalization of Roe's flux difference splitter for the euler equations , 1993 .

[18]  W. R. Briley,et al.  Solution of the three-dimensional compressible Navier-Stokes equations by an implicit technique , 1975 .

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

[20]  D. Whitfield,et al.  Discretized Newton-relaxation solution of high resolution flux-difference split schemes , 1991 .

[21]  Philip L. Roe,et al.  Characteristic time-stepping or local preconditioning of the Euler equations , 1991 .

[22]  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 .

[23]  E. Issman,et al.  Non-overlapping preconditioners for a parallel implicit Navier-Stokes solver , 1998, Future Gener. Comput. Syst..

[24]  T. Barth.  Aspects of Unstructured Grids and Finite-Volume Solvers for the Euler and Navier-Stokes Equations , 1994 .

[25]  Roland W. Freund,et al.  A Transpose-Free Quasi-Minimal Residual Algorithm for Non-Hermitian Linear Systems , 1993, SIAM J. Sci. Comput..

[26]  R. Ni.  A multiple grid scheme for solving the Euler equations , 1981 .