Multi-adaptive Galerkin methods for ODEs I

SIAM Journal on Scientific Computing

Volumn 24, Issue 6, 2003, Pages 1879-1902

  Logg, Anders ^a  

^a CHALMERS UNIVERSITY OF TECHNOLOGY   (Sweden)

Author keywords

Adaptivity; Continuous Galerkin; Discontinuous Galerkin; Global error control; Individual time steps; Local time steps; McG(q); MdG(q); Multi adaptivity; ODE

Indexed keywords

ADAPTIVE ALGORITHMS; BOUNDARY CONDITIONS; C (PROGRAMMING LANGUAGE); COMPUTER SIMULATION; ERROR ANALYSIS; FINITE ELEMENT METHOD; INTEGRAL EQUATIONS; ITERATIVE METHODS; ORDINARY DIFFERENTIAL EQUATIONS; PARTIAL DIFFERENTIAL EQUATIONS; POLYNOMIAL APPROXIMATION; SOLAR SYSTEM;

CONTINUOUS GALERKIN; DISCONTINUOUS GALERKIN; GLOBAL ERROR CONTROL; INDIVIDUAL TIME STEPS; LOCAL TIME STEPS; MULTI ADAPTIVE GALERKIN METHODS;

GALERKIN METHODS;

EID: 0346316950     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1064827501389722     Document Type: Article

References (34)

1. S. Alexander and C. Agnor, n-body simulations of late stage planetary formation with a simple fragmentation model, ICARUS, 132 (1998), pp. 113-124.
2. U. M. Ascher and L. R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, SIAM, Philadelphia, 1998.
3. J. Butcher, The Numerical Analysis of Ordinary Differential Equations. Runge-Kutta and General Linear Methods, John Wiley, Chichester, UK, 1987.
4. G. Dahlquist, Stability and Error Bounds in the Numerical Integration of Ordinary Differential Equations, Kungl. Tekn. Högsk Handl. Stockholm. No., 130 (1959).
5. R. Davé, J. Dubinski, and L. Hernquist, Parallel treesph, New Astronomy, 2 (1997), pp. 277-297.
6. C. Dawson and R. Kirby, High resolution schemes for conservation laws with locally varying time steps, SIAM J. Sci. Comput., 22 (2001), pp. 2256-2281.
7. M. Delfour, W. Hager, and F. Trochu, Discontinuous Galerkin methods for ordinary differential equations, Math. Comp., 36 (1981), pp. 455-473.
8. K. Eriksson, D. Estep, P. Hansbo, and C. Johnson, Introduction to adaptive methods for differential equations, in Acta Numerica, 1995, Acta Numer., Cambridge University Press, Cambridge, 1995, pp. 105-158.
9. K. Eriksson, D. Estep, P. Hansbo, and C. Johnson, Computational Differential Equations, Cambridge University Press, Cambridge, 1996.
10. K. Eriksson and C. Johnson, Adaptive Finite Element Methods for Parabolic Problems III: Time Steps Variable in Space, manuscript, Chalmers University of Technology, Göteborg, Sweden.
11. K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems I: A linear model problem, SIAM J. Numer. Anal., 28 (1991), pp. 43-77.
12. ∞, SIAM J. Numer. Anal., 32 (1995), pp. 706-740.
13. K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems IV: Nonlinear problems, SIAM J. Numer. Anal., 32 (1995), pp. 1729-1749.
14. K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems V: Long-time integration, SIAM J. Numer. Anal., 32 (1995), pp. 1750-1763.
15. K. Eriksson, C. Johnson, and S. Larsson, Adaptive finite element methods for parabolic problems VI: Analytic semigroups, SIAM J. Numer. Anal., 35 (1998), pp. 1315-1325.
16. K. Eriksson, C. Johnson, and V. Thomée, Time discretization of parabolic problems by the discontinuous Galerkin method, RAIRO Modél. Math. Anal. Numér., 19 (1985), pp. 611-643.
17. D. Estep, A posteriori error bounds and global error control for approximation of ordinary differential equations, SIAM J. Numer. Anal., 32 (1995), pp. 1-48.
18. D. Estep and D. French, Global error control for the continuous Galerkin finite element method for ordinary differential equations, RAIRO Modél. Math. Anal. Numér., 28 (1994), pp. 815-852.
19. D. Estep and R. Williams, Accurate parallel integration of large sparse systems of differential equations, Math. Models Methods Appl. Sci., 6 (1996), pp. 535-568.
20. J. Flaherty, R. Loy, M. Shephard, B. Szymanski, J. Teresco, and L. Ziantz, Adaptive local refinement with octree load balancing for the parallel solution of three-dimensional conservation laws, J. Parallel Distrib. Comput., 47 (1997), pp. 139-152.
21. E. Hairer and G. Wanner, Solving Ordinary Differential Equations. I. Nonstiff Problems, Springer Ser. Comput. Math. 8, Springer-Verlag, Berlin, 1987.
22. E. Hairer and G. Wanner, Solving Ordinary Differential Equations. II. Stiff and Differential-Algebraic Problems, Springer Ser. Comput. Math. 14, Springer-Verlag, Berlin, 1991.
23. P. Hansbo, A note on energy conservation for hamiltonian systems using continuous time finite elements, Comm. Numer. Methods Engrg., 17 (2001), pp. 863-869.
24. C. Johnson, Error estimates and adaptive time-step control for a class of one-step methods for stiff ordinary differential equations, SIAM J. Numer. Anal., 25 (1988), pp. 908-926.
25. O. Kessel-Deynet, Berücksichtigung ionisierender Strahlung im Smoothed-Particle-Hydrodynamics-Verfahren und Anwendung auf die Dynamik von Wolkenkernen im Strahlungsfeld massiver Sterne, Ph.D. thesis, Naturwissenschaftlich-Matehmatischen Gesamtfakultät, Ruprecht-Karls-Universität, Heidelberg, 1999.
26. A. Logg, Multi-Adaptive Error Control for ODES, NA Group Report 98/20, Oxford University Computing Laboratory, Oxford, UK, 1998; also available online fromhttp://www.phi.chalmers.se/preprints/abstracts/preprint-2000-03.html.
27. A. Logg, A Multi-Adaptive ODE-Solver, M.Sc. thesis, Department of Mathematics, Chalmers University of Technology, Göteborg, Sweden, 1998; also available online from http://www.phi.chalmers.se/preprints/abstracts/preprint-2000-02.html.
28. A. Logg, Multi-Adaptive Galerkin Methods for ODES I: Theory & Algorithms, Chalmers Finite Element Center Preprint 2001-09, http://www.phi.chalmers.se/preprints/abstracts/preprint-2000-09.html (25 February 2001).
29. A. Logg, Multi-adaptive Galerkin methods for ODES II: Implementation and applications, SIAM J. Sci. Comput., submitted.
30. A. Logg, Tanganyika, version 1.2.1, http://www.phi.chalmers.se/tanganyika/ (10 May 2001).
31. J. Makino and S. Aarseth, On a hermite integrator with Ahmad-Cohen scheme for gravitational many-body problems, Publ. Astron. Soc. Japan, 44 (1992), pp. 141-151.
32. K.-S. Moon, A. Szepessy, R. Tempone, and G. Zourakis, Adaptive Approximation of Differential Equations Based on Global and Local Errors, preprint. TRITA-NA-0006 NADA, KTH, Stockholm, Sweden, 2000.
33. S. Osher and R. Sanders, Numerical approximations to nonlinear conservation laws with locally varying time and space grids, Math. Comp., 41 (1983), pp. 321-336.
34. L. Shampine, Numerical Solution of Ordinary Differential Equations, Chapman and Hall, London, 1994.