Dynamical Approach Study Of Spurious Steady-State Numerical Solutions Of Nonlinear Differential Equations II. Global Asymptotic Behavior Of Time Discretiza Tions

International Journal of Computational Fluid Dynamics

Volumn 4, Issue 3-4, 1995, Pages 219-283

  Yee, H C ^a   Sweby, P K ^b  

^a NASA AMES RESEARCH CENTER   (United States)

^b UNIVERSITY OF READING   (United Kingdom)

Author keywords

global asymptotic behavior; nonlinear ODEs; numerical methods; spurious asymptotes; Spurious steady state numerical solutions; time discretizations

Indexed keywords

EID: 84954811121     PISSN: 10618562     EISSN: 10290257     Source Type: Journal    
DOI: 10.1080/10618569508904525     Document Type: Article

References (37)

1. Adams, E. (1990) “Periodic Solutions: Enclosure, Verification, and Applications”, Computer Arithmetic and Self-Validating Numerical Methods, Academic Press, 199–245.
2. Beam, R. M. and Bailey, H. E. (1988) “Direct Solver for Navier-Stokes Equations”, Proceedings of International Conference on Computational Engineering Science, Atlanta, GA.
3. Beam, R. M. and Warming, R. F. (1976) “An Implicit Finite-Difference Algorithm for Hyperbolic Systems in Conservation Law Form”, J. Comput. Phys., 22, 87–110.
4. Budd, C. J., Stuart, A. M., Koomullil, G. P. and Yee, H. C. (1994) “Numerical Solution Behavior of Model Convection-Diffusion BVP with Grid Adaptation”, in preparation.
5. Foias, C., Sell, G. and Temam, R. (1985)“Varietes Inertielles des Equations Differentielles Dissipatives”, C.R. Acad. Sci. Paris, Ser. I Math., 301, 139–141.
6. Globus, A., Levit, C. and Lasinski, T. (1991) “A Tool for Visualizing the Topology of Three-Dimensional Vector Fields”, NAS Applied Research Branch Report RNR-91-017, NASA Ames Research Center.
7. Griffiths, D. F., Sweby, P. K. and Yee, H. C. (1992a) “On Spurious Asymptotes Numerical Solutions of Explicit Runge-Kutta Schemes”, IMA J. Numer. Anal., 12, 319–338.
8. Griffiths, D. F., Stuart, A. M. and Yee, H. C. (1992b)“Numerical Wave Propagation in Hyperbolic Problems with Nonlinear Source Terms”, SIAM J. of Numer. Analy., 29(5), 1244–1260.
9. Guckenheimer, J. and Holmes, P. (1983) Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York.
10. Hale, J. and Kocak, H. (1991) Dynamics and Bifurcations, Springer-Verlag, New York.
11. Hairer, E., Iserles, A. and Sanz-Serna, J. M.(1989) “Equilibria of Runge-Kutta Methods”, Numer. Math.
12. Hsu, C. S. (1987) Cell-to-Celi Mapping, Springer-Verlag, New York.
13. Hung, C. M., Sung, C. H. and Chen, C. L. (1991) “Computation oFSaddle Point of Attachment” AIAA-91-1713, A1AA 22nd Fluid Dynamics, Plasma Dynamics and Lasers Conference, Honolulu, Hawaii.
14. Humphries, A. R. (1992) “Spurious Solutions of Numerical Methods for Initial Value problems”, IMA J. Num. Anal.
15. Iserles, A. (1988) “Stability and Dynamics of Numerical Methods for Nonlinear Ordinary Differential Equations”, DAMTP NA1, University of Cambridge, Cambridge, England.
16. Iserles, A., Peplow, A. T. and Stuart, A. M. (1990) “A Unified Approach to Spurious Solutions Introduced by Time Discretisation”, Part I: Basic Theory, DAMTP 1990/NA4, Numerical Analysis Reports, University of Cambridge.
17. Jameson, A. (1991) “Airfoils Admitting Nonunique Solutions to the Euler Equations”, A1AA-91-1625.
18. Keller, H. B. (1977) “Numerical Solution of Bifurcation and Nonliner Eigenvalue Problems”, Applications of Bifurcation Theory, P. H. Rabinowitz, ed., Academic Press, 359–384.
19. Kwak, M. (1991) “Finite Dimensional Inertial Forms for the 3D Navier-Stokes Equations,” IMA preprint Series #828.
20. Lafon, A. and Yee, H. C. (1991) “Dynamical Approach Study of Spurious Steady-State Numerical Solutions for Nonlinear Differential Equations”, Part HI: The Effects of Nonlinear Source Terms and Boundary Conditions in Reaction-Convection Equations, NASA TM-103877, to appear in Intern. J. CFD.
21. Lafon, A. and Ycc, H. C. (1992) “Dynamical Approach Study of Spurious Steady-State Numerical Solutions of Nonlinear Differential Equations”, Part IV: Stability vs. Numerical Treatment of Nonlinear Source Terms, ONER A-CERT Technical Report DERAT 45/5005.38, to appear in Intern. J. CFD.
22. Lambert, J. D. (1973) Computational Methods in Ordinary Differential Equations, John Wiely, New York.
23. Langford, W. F. and looss, G. (1980) “Interactions of Hopf and Pitchfork Bifurcations”, ISNM 54, Bifurcation Problems and Their Numerical Solution, Workshop on Bifurcation Problems and Their Numerical Solution, Dortmund, ed. H. D. Mittelmann and H. Weber, Birkhauser Verlag, Basel, 103–134.
24. Panov, A. M. (1956) “Behavior of the Trajectories of a System of Finite Difference Equations in the Neighbourhood of a Singular Point”, Uch. Zap. Urt!l. Gos. Univ. vyp, 19, 89–99.
25. Perron, O. (1929) “Uber Stabilitat und Asymtotisches Uberhalten de Losungen eines Systems endlicher Diffcrenzengleichungen”, J. Reine Anew. Math 161,41–64.
26. Richtmyer, R. D. and Morton, K. W. (1967) Difference Methods for Initial-Value Problems, Interscience-Wicly, New York.
27. Sanz-Serna, J. M. (1990) “Numerical Ordinary Differential Equations vs. Dynamical Systems,” Applied Math. Comput. Report 1990/3, Universidad de Valladolid.
28. Schecter, S. and Shearer, M. (1990) “Undercomprcssive Shocks for Nonstriclly Hyperbolic Conservation Laws”, IMA Preprint Series #619.
29. Shearer, M., Schaeffer, D. G., Marchesin, D. and Paes-Leme, P. (1987) “Solution of the Riemann Problem for a Prototype 2x2 System of Non-Strictly Hyperbolic Conservation Laws”, Arch. Rat. Mech. Anal., 97, 299–320.
30. Strang, G. (1968) “On the Construction and Comparison of Difference Schemes”, SIAM J. Num. Anal., 5, 506–517.
31. Swcby. P, K., Yee, H.C. and Griffiths, D. F. (1990) “On Spurious Steady-State Solutions of Explicit Runge-Kutta Schemes”, University of Reading, Department of Mathematics, Numerical Analysis 3/90. also NASA TM 102819, April 1990.
32. Swcby, P. K. and Yee, H. C. (1991) “On Spurious Asymptotic Numerical Solutions of 2 x 2 Systems of ODEs”, Numerical analysis Report 7/91, University of Reading, England.
33. Swcby, P. K. and Yee, H. C. (1994) “On the Dynamics of Some Grid Adaptation Schemes”, Proceedings of the 4th International Conference on Numerical Grid Generation in CFD and Related Fields,” University College of Swansea UK, also R1ACS Technical Report 94.02, Feb. 1994.
34. Tcmam, R. (1989) “Do Inertial Manifolds Apply to Turbulence?”, Physica D, 37, 146–152.
35. Warner, B. (1980) “Turning Points of Branches of Positive Solutions”, ISNM 54, Bifurcation Problems and Their Numerical Solution, Workshop on Bifurcation Problems and Their Numerical Solution, Dortmund, ed. H. D. Mittelmann and H. Weber, Birkhauser Verlag, Basel, 211–226.
36. Ycc, H.C. (1989) “A Class of High-Rcsolution Explicit and Implicit Shock-Capturing Methods”, VKI Lecture Series 1989–04 March 6–10, 1989, also NASA TM-101088, Feb. 1989.
37. Ycc, H. C. (1991) “A Nonlinear Dynamical Approach to Algorithm Development in Hypersonic CFD”, Proceedings of the4lh International symposium on Computational Fluid Dynamics, Davis, Calif.