Increment formulations for rounding error reduction in the numerical solution of structured differential systems

Future Generation Computer Systems

Volumn 19, Issue 3, 2003, Pages 375-383

  Sofroniou, Mark ^a   Spaletta, Giulia ^b  

^a WOLFRAM RESEARCH INC   (United States)

^b UNIVERSITY OF BOLOGNA   (Italy)

Author keywords

Composition methods; Geometric numerical integration; Orthogonal projection; Separable Hamiltonian differential equations; Symplectic methods

Indexed keywords

ALGORITHMS; CHEBYSHEV APPROXIMATION; HAMILTONIANS; INTEGRATION; PARTIAL DIFFERENTIAL EQUATIONS; PROBLEM SOLVING; RUNGE KUTTA METHODS;

INCREMENT FORMATION;

ORDINARY DIFFERENTIAL EQUATIONS;

EID: 0037399925     PISSN: 0167739X     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0167-739X(02)00164-4     Document Type: Article

References (36)

1. A. Abdulle, Chebyshev methods based on orthogonal polynomials, Ph.D. Thesis, Section de Mathématiques, Université de Genève, 2001.
2. E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, D. Sorenson, LAPACK Users' Guide, 3rd ed., SIAM, Philadelphia, 1999.
3. J.C. Butcher, The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods, Wiley, Chichester, 1987.
4. Butcher J.C., Order, stepsize and stiffness switching. Computing. 44:1990;209-220.
5. Del Buono N., Lopez L., Runge-Kutta type methods based on geodesics for systems of ODEs on the Stiefel manifold. BIT. 41(5):2001;912-923.
6. Dieci L., Russel R.D., Van Vleck E.S., Unitary integrators and applications to continuous orthonormalization techniques. SIAM J. Numer. Anal. 31:1994;261-281.
7. Dieci L., Van Vleck E.S., Computation of a few Lyapunov exponents for continuous and discrete dynamical systems. Appl. Numer. Math. 17(3):1995;275-291.
8. Dieci L., Van Vleck E.S., Computation of orthonormal factors for fundamental solution matrices. Numer. Math. 83:1999;591-620.
9. Earn D.J.D., Tremaine S., Exact numerical studies of Hamiltonian maps: iterating without roundoff error. Physica D. 56:1992;1-22.
10. Gill S., A process for the step-by-step integration of differential equations in an automatic digital computing machine. Proc. Camb. Phil. Soc. 47:1951;96-108.
11. Gladman B., Duncan M., Candy J., Symplectic integrators for long-term integrations in celestial mechanics. Celest. Mech. 52:1991;221-240.
12. Gladwell I., Shampine L.F., Brankin R.W., Automatic selection of the initial step size for an ODE solver. J. Comp. Appl. Math. 18:1987;175-192.
13. G.H. Golub, C.F. Van Loan, Matrix Computations, 3rd ed., Johns Hopkins University Press, Baltimore, MD, 1996.
14. E. Hairer, C. Lubich, G. Wanner, Geometric numerical integration: structure-preserving algorithms for ordinary differential equations, Springer Ser. Comput. Math. 31, Springer, New York, 2002.
15. E. Hairer, S.P. Nørsett, G. Wanner, Solving Ordinary Differential Equations. I. Nonstiff Problems, 2nd ed., Springer, New York, 1993.
16. E. Hairer, G. Wanner, Solving Ordinary Differential Equations. II. Stiff and Differential Algebraic Problems, 2nd ed., Springer, New York, 1996.
17. Higham D., Time-stepping and preserving orthonormality. BIT. 37(1):1997;236-241.
18. N.J. Higham, Matrix nearness problems and applications, in: M.J.C. Gover, S. Barnett (Eds.), Applications of Matrix Theory, Oxford University Press, Oxford, 1989, pp. 1-27.
19. N.J. Higham, Accuracy and Stability of Numerical Algorithms, SIAM, Philadelphia, 1996.
20. Kahan W.H., Li R.C., Composition constants for raising the order of unconventional schemes for ordinary differential equations. Math. Comp. 66:1997;1089-1099.
21. I. Koren, Computer Arithmetic Algorithms, Prentice-Hall, Englewood Cliffs, NJ, 1993.
22. Iserles A., Munthe-Kaas H.Z., Nørsett S.P., Zanna A., Lie-group methods. Acta Numer. 9:2000;215-365.
23. McLachlan R.I., On the numerical integration of ordinary differential equations by symmetric composition methods. SIAM J. Sci. Comp. 16:1995;151-168.
24. McLachlan R.I., Atela P., The accuracy of symplectic integrators. Nonlinearity. 5:1992;541-562.
25. R.I. McLachlan, G.R.W. Quispel, Six lectures on the geometric integration of ODEs, in: R.A. DeVore, A. Iserles, E. Süli (Eds.), Foundations of Computational Mathematics, Cambridge University Press, Cambridge, 2001, pp. 155-210.
26. R.I. McLachlan, G.R.W. Quispel, Splitting methods, Acta Numerica 11, Cambridge University Press, Cambridge (2002) 341-434.
27. Murua A., On order conditions for partitioned symplectic methods. SIAM J. Numer. Anal. 34(6):1997;2204-2211.
28. Okunbor D.I., Skeel R.D., Explicit canonical methods for Hamiltonian systems. Math. Comp. 59:1992;439-455.
29. Quinn T., Tremaine S., Roundoff error in long-term planetary orbit integrations. Astron. J. 99(3):1990;1016-1023.
30. J.M. Sanz-Serna, M.P. Calvo, Numerical Hamiltonian Problems, Chapman & Hall, London, 1994.
31. M. Sofroniou, G. Spaletta, Solving orthogonal matrix differential systems in Mathematica, in: P.M.A. Sloot, C.J.K. Tan, J.J. Dongarra, A.G. Hoekstra (Eds.), Computational Science ICCS 2002, Lecture Notes in Computer Science, vol. 2330, Springer, Berlin, 2002, pp. 496-505.
32. M. Sofroniou, G. Spaletta, Symplectic methods for separable Hamiltonian systems, in: P.M.A. Sloot, C.J.K. Tan, J.J. Dongarra, A.G. Hoekstra (Eds.), Computational Science ICCS 2002. Lecture Notes in Computer Science, vol. 2330, Springer, Berlin, 2002, pp. 506-515.
33. Suzuki M., Fractal decomposition of exponential operators with applications to many-body theories and Monte Carlo simulations. Phys. Lett. A. 146:1990;319-323.
34. Suzuki M., General theory of higher-order decomposition of exponential operators and symplectic integrators. Phys. Lett. A. 165:1992;387-395.
35. R.C. Whaley, A. Petitet, J.J. Dongarra, Automated empirical optimization of software and the ATLAS (automatically tuned linear algebra software) project. http://math-atlas.sourceforge.net.
36. Yoshida H., Construction of high order symplectic integrators. Phys. Lett. A. 150:1990;262-268.