On balanced approximations for time integration of multiple time scale systems

Journal of Computational Physics

Volumn 185, Issue 2, 2003, Pages 583-611

  Knoll, D A ^a   Chacon, L ^a   Margolin, L G ^b   Mousseau, V A ^a  

^a LOS ALAMOS NATIONAL LABORATORY   (United States)

^b LOS ALAMOS NATIONAL LABORATORY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

DIFFERENCE EQUATIONS; FLOW OF WATER; MAGNETOHYDRODYNAMICS; NONLINEAR EQUATIONS; ORDINARY DIFFERENTIAL EQUATIONS; PARTIAL DIFFERENTIAL EQUATIONS; TIME MEASUREMENT;

LINEAR AND NONLINEAR PROBLEMS; LINEARISATION; MODIFIED EQUATION ANALYSIS; MULTIPLE TIME SCALE; NONLINEAR ORDINARY DIFFERENTIAL EQUATION; NUMERICAL APPROXIMATIONS; SCALE SYSTEM; SIMPLE++; SPLITTINGS; TIME-INTEGRATION;

LINEARIZATION;

EID: 0037336326     PISSN: 00219991     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0021-9991(03)00008-1     Document Type: Article

References (34)

1. J. Brackbill, B. Cohen, Multiple Time Scales, Academic Press, Orlando, 1985.
2. P.N. Brown, Y. Saad, Hybrid Krylov methods for nonlinear systems of equations, SIAM J. Sci. Stat. Comput. 11 (1990) 450-481.
3. T.F. Chan, K.R. Jackson, Nonlinearly preconditioned Krylov subspace methods for discrete Newton algorithms, SIAM J. Sci. Stat. Comput. 5 (1984) 533-542.
4. G. Strang, On the construction and comparison of difference schemes, SIAM J. Numer. Anal. 5 (1968) 506-517.
5. G. Browning, H.-O. Kriess, Splitting methods for problems with different time scales, Mon. Weather Rev. 122 (1994) 2614-2622.
6. B. Sportisse, An analysis of operator splitting techniques in the stiff case, J. Comp. Phys. 161 (2000) 140-168.
7. C. Hirt, Heuristic stability theory for finite difference equations, J. Comp. Phys. 2 (1968) 339-355.
8. R. Warming, B. Hyett, The modified equation approach to the stability and accuracy analysis of finite-difference methods, J. Comp. Phys. 14 (1974) 159-179.
9. D.F. Griffiths, J. Sanz-Serna, On the scope of the method of modified equations, SIAM J. Sci. Stat. Comput. 3 (1986) 994-1008.
10. J.N. Shadid, C.C. Ober, R.P. Pawlowski, D.L. Ropp, Studies on Solution Methods for Nonlinear, Multiple-Time-Scale, PDE Simulations: examples from diffusion reaction systems, in: Proc. of Seventh Copper Mountain Conference on Iterative Methods, Copper Mountain, Colorado, March 2002.
11. S.-C. Chang, A critical analysis of the modified equation technique of Warming and Hyett, J. Comp. Phys. 86 (1990) 107-126.
12. P.K. Smolarkiewicz, L.G. Margolin, MPDATA: a finite difference solver for geophysical flows, J. Comp. Phys. 140 (1998) 459-480.
13. P.K. Smolarkiewicz, L.G. Margolin, On forward-in-time differencing for fluids: extension to a curvilinear framwork, Mon. Weather Rev. 121 (1993) 1847-1859.
14. R.B. Lowrie, J.E. Morel, Methods for hyperbolic systems with stiff relaxation, Int. J. Numer. Methods Fluids 40 (2002) 413-423.
15. D.A. Jones, L.G. Margolin, E.S. Titi, On the effectiveness of the approximate inertial manifold, Theor. Comp. Fluid Dyn. 7 (1995) 243-260.
16. D.A. Jones, L.G. Margolin, A.C. Poje, Enslaved finite difference schemes for nonlinear dissipative PDEs, Numer. Methods PDEs 12 (1996) 13-40.
17. A.C. Poje, D.A. Jones, L.G. Margolin, Enslaved finite difference approximations for quasi-geostrophic shallow flows, Physica D 98 (1996) 559-573.
18. Y. Saad, Iterative Methods for Sparse Linear Systems, PWS Publishing Company, Boston, 1996.
19. D.A. Knoll, W.J. Rider, G.L. Olson, Nonlinear convergence, accuracy, and time step control in non-equilibrium radiation diffusion, J. Quant. Spectrosc. Radiat. Transfer 70 (2001) 25-36.
20. L. Chacon, D.A. Knoll, J.M. Finn, An implicit nonlinear reduced resistive MHD solver, J. Comput. Phys. 178 (2002) 15-36.
21. V. Mousseau, D.A. Knoll, J. Reisner, An implicit nonlinearly consistent method for the two-dimensional shallow-water equations with Coriolis force, Mon. Weather Rev. 130 (2002) 2611-2625.
22. R.L. Bowers, J.R. Wilson, Numerical Modeling in Applied Physics and Astrophysics, Jones and Bartlett, Boston Publishers, 1991.
23. E.S. Oran, J.P. Boris, Numerical Simulation of Reactive Flow, Elsevier, New York, 1987.
24. C. Baldwin, P.N. Brown, R. Falgout, F. Graziani, J. Jones, Iterative linear solvers in a 2d radiation-hydrodynamics code: methods and performance, J. Comput. Phys. 154 (1999) 1-40.
25. A.M. Winslow, Multifrequency-gray method for radiation diffusion with Compton scattering, J. Comput. Phys. 117 (1995) 262-273.
26. D.A. Knoll, W.J. Rider, G.L. Olson, An efficient nonlinear solution method for non-equilibrium radiation diffusion, J. Quant. Spectrosc. Radiat. Transfer 63 (1999) 15-29.
27. W.J. Rider, D.A. Knoll, Time step size selection for radiation diffusion calculations, J. Comput. Phys. 152 (1999) 790-795.
28. F.A. Williams, Combustion Theory, Addison-Wesley Publishers, Reading, Mass, 1985.
29. D. Biskamp, Nonlinear Magnetohydrodynamics, Cambridge University Press, Cambridge, 1993.
30. H.R. Strauss, Nonlinear, 3-dimensional magnetohydrodynamics of noncircular tokamaks, Phys. Fluids 19 (1) (1976) 134-140.
31. J.F. Drake, T.M. Antonsen, Nonlinear reduced fluid equations for toroidal plasmas, Phys. Fluids 27 (4) (1984) 898-908.
32. R.D. Hazeltine, M. Kotschenreuther, P.J. Morrison, A four-field model for tokamak plasma dynamics, Phys. Fluids 28 (8) (1985) 2466-2477.
33. H.P. Furth, J. Killeen, M.N. Rosenbluth, Finite-resistivity instabilities of a sheet pinch, Phys. Fluids 4 (1963) 459-484.
34. D. Harned, W. Kerner, Semi-implicit method for three-dimensional compressible magnetohydrodynamic simulation, J. Comp. Phys. 60 (1985) 62-75.