Triangularly implicit iteration methods for ODE-IVP solvers

SIAM Journal on Scientific Computing

Volumn 18, Issue 1, 1997, Pages 41-55

  Van Der Houwen, P J ^a   De Swart, J J B ^a  

^a CWI   (Netherlands)

Author keywords

Convergence of iteration methods; Numerical analysis; Parallelism; Runge Kutta methods

Indexed keywords

BLACK DIAGONAL NEWTON JACOBIAN APPROXIMATION; BLOCK TRIANGULAR APPROXIMATION; TRIANGULARLY IMPLICIT ITERATION METHOD;

APPROXIMATION THEORY; CONVERGENCE OF NUMERICAL METHODS; INITIAL VALUE PROBLEMS; ITERATIVE METHODS; NUMERICAL ANALYSIS; PARALLEL PROCESSING SYSTEMS; RUNGE KUTTA METHODS;

ORDINARY DIFFERENTIAL EQUATIONS;

EID: 0030621829     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1064827595287456     Document Type: Article

References (23)

1. A. BELLEN, Parallelism across the steps for difference and differential equations, Lecture Notes in Mathematics 1386, Springer-Verlag, Berlin, 1987, pp. 22-35.
2. A. BELLEN, Z. JACKIEWICZ, AND M. ZENNARO, Time-point relaxation Runge-Kutta methods for ordinary differential equations, J. Comput. Appl. Math., 45 (1990), pp. 121-138.
3. K. BURRAGE, A special family of Runge-Kutta methods for solving stiff differential equations, BIT, 18 (1978), pp. 373-383.
4. K. BURRAGE, Parallel methods for initial value problems, Appl. Numer. Math., 11 (1993), pp. 5-26.
5. K. BURRAGE, The search for the Holy Grail, or: Predictor-corrector methods for solving ODEIVPs, Appl. Numer. Math., 11 (1993), pp. 125-141.
6. J. C. BUTCHER, On the implementation of implicit Runge-Kutta methods, BIT, 16 (1976), pp. 237-240.
7. J. C. BUTCHER, The Numerical Analysis of Ordinary Differential Equations, Runge-Kutta and General Linear Methods, Wiley, New York, 1987.
8. P. CHARTIER, Parallelism in the Numerical Solutions of Initial Value Problems for ODEs and DAEs, Ph.D. thesis, Université de Rennes I, France, 1993.
9. W. H. ENRIGHT, T. E. HULL, AND B. LINDBERG, Comparing numerical methods for stiff systems of ODEs, BIT, 15 (1975), pp. 10-48.
10. C. W. GEAR, The automatic integration of stiff ordinary differential equations, in Proc. IFIP Congress 1968, North-Holland, Amsterdam, 1969, pp. 187-193.
11. C. W. GEAR AND X. XUHAI, Parallelism across time in ODEs, Appl. Numer. Math., 11 (1993), pp. 45-68.
12. G. H. GOLUB AND C. F. VAN LOAN, Matrix Computations, 2nd ed., John Hopkins University Press, Baltimore, MD, London, 1989.
13. B. A. GOTTWALD, MISS - ein einfaches simulations-system für biologische und chemische prozesse, EDV in Medizin und Biologie, 3 (1977), pp. 85-90.
14. E. HAIRER AND G. WANNER, Solving Ordinary Differential Equations II: Stiff and Differentialalgebraic Problems, Springer-Verlag, Berlin, 1991.
15. P. J. VAN DER HOUWEN and B. P. SOMMEIJER, Iterated Runge-Kutta methods on parallel computers, SIAM J. Sci. Statist. Comput., 12 (1991), pp. 1000-1028.
16. P. J. VAN DER HOUWEN, B. P. SOMMEIJER, AND W. A. VAN DER VEEN, Parallelism across the steps in iterated Runge-Kutta methods for stiff initial value problems, Numer. Algorithms, 8 (1994), pp. 293-312.
17. P. J. VAN DER HOUWEN, B. P. SOMMEIJER, AND W. A. VAN DER VEEN, Parallel iteration across the steps of high order Runge-Kutta methods for nonstiff initial value problems, J. Comput. Appl. Math., 60 (1995), pp. 309-329.
18. W. M. LIOEN, On the Diagonal Approximation of Full Matrices, Tech. Report NM-R9518, CWI, Amsterdam, the Netherlands, 1995, J. Comput. Appl. Math., to appear.
19. W. L. MIRANKER AND W. LINIGER, Parallel methods for the numerical integration of ordinary differential equations, Math. Comp., 21 (1967), pp. 303-320.
20. B. OREL, Parallel Runge-Kutta methods with real eigenvalues, Appl. Numer. Math., 11 (1993), pp. 241-250.
21. B. P. SOMMEIJER, Parallelism in the numerical integration of initial value problems, CWI Tract 99, CWI, Amsterdam, the Netherlands, 1993.
22. W. M. LIOEN, J. J. B. DE SWART, AND W. A. VAN DER VEEN, Test set for IVP solvers. Available via WWW at URL http://www.cwi.nl/cwi/projects/IVPtestset.shtml, 1996.
23. J. G. VERWER, Gauss-Seidel iteration for stiff ODEs from chemical kinetics, SIAM J. Sci. Comput., 15 (1994), pp. 1243-1259.