Implicit Solvers for Unstructured Meshes

Implicit methods are developed and tested for unstructured mesh computations. The methods are used to solve the compressible Navier-Stokes equations to steady state. The approximate system which arises from the Newton linearization of the nonlinear evolution operator is solved by using the preconditioned GMRES (generalized minimum residual) technique. Three different preconditioners, namely, the incomplete LU factorization (ILU), block diagonal factorization, and the symmetric successive over-relaxation (SSOR) are investigated. The preconditioners have been optimized to have good vectorization properties. SSOR and ILU themselves are studied as iterative schemes. The various methods are compared over a wide range of problems. Ordering of the unknowns, which affects the convergence of these sparse matrix iterative methods, is also investigated. Results are presented for inviscid and turbulent viscous calculations on single and multi-element airfoil configurations using globally and adaptively generated meshes.

[1]  Joel H. Saltz,et al.  Automated problem scheduling and reduction of synchronization delay effects , 1987 .

[2]  I. Duff,et al.  Direct Methods for Sparse Matrices , 1987 .

[3]  A Jameson,et al.  CALCULATION OF IN VISCID TRANSONIC FLOW OVER A COMPLETE AIRCRAFT , 1986 .

[4]  Joel H. Saltz,et al.  Aggregation Methods for Solving Sparse Triangular Systems on Multiprocessors , 1990, SIAM J. Sci. Comput..

[5]  Jaime Peraire,et al.  An implicit finite element method for high speed flows , 1991 .

[6]  Dimitri J. Mavriplis,et al.  Algebraic turbulence modeling for unstructured and adaptive meshes , 1990 .

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

[8]  D. Mavriplis.  Multigrid solution of the two-dimensional Euler equations on unstructured triangular meshes , 1987 .

[9]  Yousef Saad,et al.  Solving Sparse Triangular Linear Systems on Parallel Computers , 1989, Int. J. High Speed Comput..

[10]  Timothy J. Barth,et al.  The design and application of upwind schemes on unstructured meshes , 1989 .

[11]  Jacques Periaux,et al.  Numerical simulation of 3-D hypersonic Euler flows around space vehicles using adapted finite elements , 1987 .

[12]  Timothy J. Barth,et al.  Application of direct solvers to unstructured meshes for the Euler and Navier-Stokes equations using upwind schemes , 1989 .

[13]  I. Duff,et al.  The effect of ordering on preconditioned conjugate gradients , 1989 .

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

[15]  John T. Batina,et al.  Implicit flux-split Euler schemes for unsteady aerodynamic analysis involving unstructured dynamic meshes , 1990 .

[16]  G. Golub,et al.  Gmres: a Generalized Minimum Residual Algorithm for Solving , 2022 .