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An efficient and robust implicit operator for upwind point Gauss-Seidel method. (English) Zbl 1123.76051
Summary: An efficient and robust implicit operator for the point Gauss-Seidel method is presented for solving the compressible Euler equations. The new implicit operator was derived by adding a scalar form of artificial dissipation to the upwind implicit side. The amount of artificial dissipation was locally adjusted using a weighting factor based on the solution gradient. For validation, the performance of the new implicit operator was compared in detail with that of several existing implicit operators which have been widely used for solving the flow equations. Numerical experiments showed that the stability and convergence characteristics of the new implicit operator are significantly better than those of other existing implicit operators for calculating flows ranging from subsonic to hypersonic speeds.

76M25 Other numerical methods (fluid mechanics) (MSC2010)
65F10 Iterative numerical methods for linear systems
Full Text: DOI
[1] T.J. Barth, Analysis of implicit local linearization techniques for upwind and TVD algorithms, AIAA Paper87-0595, 1987.
[2] A. Jameson, W. Schmidt, E. Turkel, Numerical solutions of the Euler equations by finite volume methods using Runge-Kutta time-stepping schemes, AIAA Paper 81-1259, 1981.
[3] Pulliam, T.H.; Steger, J.L., Implicit finite-difference simulations of three-dimensional compressible flow, Aiaa j., 18, 2, 159-167, (1978) · Zbl 0417.76039
[4] Jameson, A.; Yoon, S., Multigrid solution of the Euler equations using implicit schemes, Aiaa j., 24, 11, 1737-1743, (1986)
[5] Pulliam, T.H., Artificial dissipation models for the Euler equations, Aiaa j., 24, 12, 1931-1940, (1986) · Zbl 0611.76075
[6] M.S. Liou, B. van Leer, Choice of implicit and explicit operators for the upwind differencing method, AIAA Paper 88-0624, 1988. · Zbl 0850.76426
[7] Amaladas, J.R.; Kamath, H., Implicit and multigrid procedures for steady-state computations with upwind algorithms, Comput. fluids, 28, 187-212, (1999) · Zbl 0968.76047
[8] Beam, R.M.; Warming, R.F., An implicit factored scheme for the compressible navier – stokes equations, Aiaa j., 16, 4, 393-402, (1978) · Zbl 0374.76025
[9] S.R. Chakravarthy, Relaxation methods for unfactored implicit upwind schemes, AIAA Paper 84-0165, 1984.
[10] Saad, Y.; Schultz, M.H., GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, J. sci. stat. comput., 7, 3, 856-869, (1986) · Zbl 0599.65018
[11] Anderson, W.K.; Bonhaus, D.L., An implicit upwind algorithm for computing turbulent flows on unstructured grids, Comput. fluids, 23, 1, 1-21, (1994) · Zbl 0806.76053
[12] N.T. Frink, Assessment of an unstructured-grid method for predicting 3-d turbulent viscous flows, AIAA Paper 96-0292, 1996.
[13] Oh, W.S.; Kim, J.S.; Kwon, O.J., Numerical simulation of two-dimensional blade – vortex interactions using unstructured adaptive meshes, Aiaa j., 40, 3, 474-480, (2002)
[14] Yoon, S.; Jameson, A., Lower-upper symmetric-gauss – seidel method for the Euler and navier – stokes equations, Aiaa j., 26, 9, 1025-1026, (1988)
[15] Wright, M.J.; Candler, G.V.; Prampolini, M., Data-parallel lower – upper relaxation method for the navier – stokes equations, Aiaa j., 34, 7, 1371-1377, (1996) · Zbl 0902.76084
[16] Chen, R.F.; Wang, Z.J., Fast, block lower – upper symmetric gauss – seidel scheme for arbitrary grids, Aiaa j., 38, 12, 2238-2245, (2000)
[17] D.L. Whitfield, L.K. Taylor, Discretized Newton-relaxation solution of high resolution flux-difference split schemes, AIAA Paper 91-1539, 1991.
[18] Koren, B., Defect correction and multigrid for an efficient and accurate computation of airfoil flows, J. comput. phys., 77, 183-206, (1988) · Zbl 0642.76077
[19] Anderson, W.K.; Thomas, J.L.; van Leer, B., Comparison of finite volume flux vector splittings for the Euler equations, Aiaa j., 24, 9, 1453-1460, (1986)
[20] Venkatakrishnan, V., Convergence to steady state solutions of the Euler equations on unstructured grids with limiters, J. comput. phys., 118, 120-130, (1995) · Zbl 0858.76058
[21] Buelow, P.E.O.; Venkateswaran, S.; Merkle, C.L., Stability and convergence analysis of implicit upwind schemes, Comput. fluids, 30, 961-988, (2001) · Zbl 1032.76047
[22] R.W. MacCormack, T.H. Pulliam, Assessment of a new numerical procedure for fluid dynamics, AIAA Paper 98-2821, 1998.
[23] Taylor, A.C.; Ng, W.-F.; Walters, R.W., Upwind relaxation methods for the navier – stokes equations using inner iterations, J. comput. phys., 99, 68-78, (1992) · Zbl 0741.76053
[24] L. Mottura, L. Vigevano, M. Zaccanti, Factorized implicit upwind methods applied to inviscid flows at high Mach number, AIAA Paper 98-2822, 1998.
[25] Jameson, A.; Turkel, E., Implicit schemes and LU decompositions, Math. comput., 37, 156, 385-397, (1981) · Zbl 0533.65060
[26] Roe, P.L., Approximate Riemann solvers, parameter vectors, and difference schemes, J. comput. phys., 43, 357-372, (1981) · Zbl 0474.65066
[27] Luo, H.; Baum, J.D.; Löhner, R., A fast, matrix-free implicit method for compressible flows on unstructured grids, J. comput. phys., 146, 664-690, (1998) · Zbl 0931.76045
[28] Darmofal, D.L.; Sui, K., A robust multigrid algorithm for the Euler equations with local preconditioning and semi-coarsening, J. comput. phys., 151, 728-756, (1999) · Zbl 0943.76053
[29] W.Z. Strang, R.F. Tomaro, M.J. Grismer, The defining methods of Cobalt60: a parallel, implicit, unstructured Euler/Navier-Stokes flow solver, AIAA Paper 99-0786, 1999.
[30] Mavriplis, D.J.; Jameson, A., Multigrid solution of the navier – stokes equations on triangular meshes, Aiaa j., 28, 8, 1415-1425, (1990)
[31] D.L. De Zeeuw, A quadtree-based adaptively-refined Cartesian-grid algorithm for solution of the Euler equations, Ph.D. Thesis, University of Michigan, 1993.
[32] van Leer, B., Flux-vector splitting for the Euler equations, Lect. notes phys., 170, 507-512, (1982)
[33] Anderson, W.K.; Rausch, R.D.; Bonhaus, D.L., Implicit/multigrid algorithms for incompressible turbulent flows on unstructured grids, J. comput. phys., 128, 391-408, (1996) · Zbl 0862.76045
[34] W.O. Valarezo, C.J. Dominik, R.J. McGhee, W.L. Goodman, K.B. Paschal, Multi-element airfoil optimization for maximum lift at high Reynolds numbers, AIAA Paper 91-3332, 1991.
[35] Yee, H.C.; Klopfer, G.H.; Montagne, J.-L., High-resolution shock-capturing schemes for inviscid and viscous hypersonic flows, J. comput. phys., 88, 31-61, (1990) · Zbl 0697.76079
[36] A.N. Lyubimov, V.V. Rusanov, Gas flows past blunt bodies, NASA TT-F 715, 1973.
[37] Billing, F.S., Shock-wave shapes around spherical- and cylindrical-nosed bodies, J. spacecraft, 4, 6, 822-823, (1967)
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