Convergence analysis of pseudo-transient continuation

SIAM Journal on Numerical Analysis

Volumn 35, Issue 2, 1998, Pages 508-523

  Kelley, C T ^a   Keyes, David E ^b,c  

^a NORTH CAROLINA STATE UNIVERSITY   (United States)

^b Old Dominion University ^*  (United States)

^c NASA LANGLEY RESEARCH CENTER   (United States)

Author keywords

Global convergence; Nonlinear equations; Pseudo transient continuation; Steady state solutions

Indexed keywords

EID: 0000111761     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142996304796     Document Type: Article

References (34)

1. K. E. BRENAN, S. L. CAMPBELL, AND L. R. PETZOLD, The Numerical Solution of Initial Value Problems in Differential-Algebraic Equations, Classics in Applied Mathematics #14, SIAM, Philadelphia, PA, 1990.
2. R. BRENT, Some efficient algorithms for solving systems of nonlinear equations, SIAM J. Numer. Anal., 10 (1973), pp. 327-344.
3. P. N. BROWN, G. D. BYRNE, AND A. C. HINDMARSH, VODE: A variable coefficient ODE solver, SIAM J. Sci. Statist. Comput., 10 (1989), pp. 1038-1051.
4. P. N. BROWN AND A. C. HINDMARSH, Matrix-free methods for stiff systems of ODE's, SIAM J. Numer. Anal., 23 (1986), pp. 610-638.
5. P. N. BROWN AND A. C. HINDMARSH. Reduced storage matrix methods in stiff ODE systems, J. Appl. Math. Comp., 31 (1989), pp. 40-91.
6. P. N. BROWN, A. C. HINDMARSH, AND L. R. PETZOLD, Using Krylov methods in the solution of large-scale differential-algebraic systems, SIAM J. Sci. Comput., 15 (1994), pp. 1467-1488.
7. P. N. BROWN, A. C. HINDMARSH, AND L. R. PETZOLD, Consistent Initial Condition Calculation for Differential-Algebraic Systems, Tech. report UCRL-JC-122175, Lawrence Livermore National Laboratory, Livermore, CA. August 1995.
8. P. N. BROWN AND Y. SAAD, Hybrid Krylov methods for nonlinear systems of equations, SIAM J. Sci. Statist. Comput., 11 (1990), pp. 450-481.
9. G. D. BYRNE, Pragmatic experiments with Krylov methods in the stiff ODE setting, in Computational Ordinary Differential Equations, J. Cash and I. Duff, eds., Oxford University Press, London, 1992.
10. X. D. CAI, D. E. KEYES, AND V. VENKATAKRISHNAN, Newton-Krylov-Schwarz: An implicit solver for CFD, in Proceedings of the 8th International Conference on Domain Decomposition Methods, Domain Decomposition Methods in Sciences and Engineering, R. Glowinski, J. Periaux, Z. Shi, and O. Widlund, eds., Wiley, Chichester, 1996, pp. 387-402.
11. R. DEMBO, S. EISENSTAT, AND T. STEIHAUG, Inexact Newton methods, SIAM J. Numer. Anal., 19 (1982), pp. 400-408.
12. J. E. DENNIS AND R. B. SCHNABEL, Numerical Methods for Nonlinear Equations and Unconstrained Optimization, Prentice-Hall, Englewood Cliffs, NJ. 1983.
13. A. ERN, V. GIOVANGIGLI, D. E. KEYES, AND M. D. SMOOKE, Towards polyalgorithmic linear system solvers for nonlinear elliptic problems, SIAM J. Sci. Comput., 15 (1994), pp. 681-703.
14. C. W. GEAR, Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-Hall, Englewood Cliffs, NJ, 1971.
15. E. ISAACSON AND H. B. KELLER, Analysis of Numerical Methods, Wiley, New York, 1966.
16. H. JIANG AND P. A. FORSYTH, Robust linear and nonlinear strategies for solution of the transonic Euler equations, Comput. Fluids, 24 (1995), pp. 753-770.
17. C. T. KELLEY, Iterative Methods for Linear and Nonlinear Equations, in Frontiers in Applied Mathematics 16, SIAM, Philadelphia, PA, 1995.
18. C. T. KELLEY, C. T. MILLER, AND M. D. TOCCI, Termination of Newton/Chord Iterations and the Method of Lines, Tech. report CRSC-TR96-19, Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC. May 1996; SIAM J. Sci. Comput., to appear.
19. C. T. KELLEY, C. T. MILLER, AND M. D. TOCCI, Termination of Newton/Chord Iterations and the Method of Lines, Tech. report CRSC-TR96-19, Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC. May 1996; SIAM J. Sci. Comput., to appear.
20. D. E. KEYES, Aerodynamic applications of Newton-Krylov-Schwarz solvers, in Proc. 14th Internat. Conference on Numerical Methods in Fluid Dynamics, M. Deshpande, S. Desai, and R. Narasimha, eds., Springer, 1995, pp. 1-20.
21. D. E. KEYES AND M. D. SMOOKE, A parallelized elliptic solver for reacting flows, in Parallel Computations and Their Impact on Mechanics, A. K. Noor, ed., American Society of Mechanical Engineers, 1987, pp. 375-402.
22. W. MULDER AND B. V. LEER, Experiments with implicit upwind methods for the Euler equations, J. Comput. Phys., 59 (1985), pp. 232-246.
23. O. NEVANLINNA, Convergence of Iterations for Linear Equations, Birkhäuser, Basel, 1993.
24. E. J. NIELSEN, R. W. WALTERS, W. K. ANDERSON, AND D. E. KEYES, Application of NewtonKrylov Methodology to a Three-Dimensional Unstructured Euler Code, AIAA Paper 95-1733, June 1995.
25. P. D. ORKWIS AND D. S. MCRAE, A Newton's Method Solver for the Navier-Stokes Equations, AIAA Paper 90-1524, June 1990.
26. J. M. ORTEGA AND W. C. RHEINBOLDT, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.
27. L. R. PETZOLD, A description of DASSL: A differential/algebraic system solver, in Scientific Computing, R. S. Stepleman et al., eds., North Holland, Amsterdam, 1983, pp. 65-68.
28. K. RADHAKRISHNAN AND A. C. HINDMARSH, Description and Use of LSODE, the Livermore Solver for Ordinary Differential Equations, Tech. report URCL-ID-113855, Lawrence Livermore National Laboratory, December 1993.
29. R. SCHRIBER AND H. B. KELLER, Driven cavity flows by efficient numerical techniques, J. Comput. Phys., 49 (1983), pp. 310-333.
30. V. E. SHAMANSKII, A modification of Newton's method, Ukrainn. Mat. Zh., 19 (1967), pp. 133-138 (in Russian).
31. L. F. SHAMPINE, Implementation of implicit formulas for the solution of ODEs, SIAM J. Sci. Statist. Comput., 1 (1980), pp. 103-118.
32. M. D. SMOOKE, An error estimate for the modified Newton method with applications to the solution of nonlinear two-point boundary value problems, J. Optim. Theory Appl., 39 (1983), pp. 489-511.
33. M. D. TOCCI, C. T. KELLEY, AND C. T. MILLER, Accurate and economical solution of the pressure head form of Richards' equation by the method of lines, Adv. Water Resources, 20 (1997), pp. 1-14.
34. V. VENKATAKRISHNAN, Newton solution of inviscid and viscous problems, AIAA J., 27 (1989), pp. 885-891.