Document Zbl 0926.65073

Positivity of Runge-Kutta and diagonally split Runge-Kutta methods. (English) Zbl 0926.65073

Appl. Numer. Math. 28, No. 2-4, 309-326 (1998).

The author investigates positivity of general Runge-Kutta and diagonally split Runge-Kutta methods for the numerical solution of positive initial value problems for ordinary differential equations. Conditions for the maximal stepsize in term of the radius of positivity of the Runge-Kutta method which guarantees positivity are given. The positivity thresholds for some irreducible Runge-Kutta methods for dissipative problems are computed.

Reviewer: Iulian Coroian (Baia Mare)

Cited in 42 Documents

MSC:

65L06	Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
34A34	Nonlinear ordinary differential equations and systems
65L05	Numerical methods for initial value problems involving ordinary differential equations
65L50	Mesh generation, refinement, and adaptive methods for ordinary differential equations

Keywords:

irreducible Runge-Kutta methods; diagonally split Runge-Kutta methods; positivity of Runge-Kutta methods; maximal stepsize

Cite Review PDF

Full Text: DOI

References:

[1]	Bellen, A.; Jackiewicz, Z.; Zennaro, M., Contractivity of waveform relaxation Runge-Kutta iterations and related limit methods for dissipative systems in the maximum norm, SIAM J. Numer. Anal., 31, 499-523 (1994) · Zbl 0818.65067
[2]	Bellen, A.; Torelli, L., Unconditional contractivity in the maximum norm of diagonally split Runge-Kutta methods, SIAM J. Numer. Anal., 34, 528-543 (1997) · Zbl 0873.65074
[3]	Bolley, C.; Crouzeix, M., Conservation de la positivité lors de la discrétisation des problèmes d’évolution paraboliques, RAIRO Analyse Numérique, 12, 3, 237-245 (1978) · Zbl 0392.65042
[4]	Butcher, J. C., The Numerical Analysis of Ordinary Differential Equations—Runge-Kutta and General Linear Methods (1987), Wiley: Wiley Chichester · Zbl 0616.65072
[5]	Coppel, W. A., Stability and Asymptotic Behaviour of Differential Equations (1965), D.C. Heath and Company: D.C. Heath and Company Boston · Zbl 0154.09301
[6]	Dekker, K.; Verwer, J. G., Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations (1984), North-Holland: North-Holland Amsterdam · Zbl 0571.65057
[7]	Hairer, E.; Wanner, G., Solving Ordinary Differential Equations II (1991), Springer: Springer Berlin · Zbl 0729.65051
[8]	Horváth, Z., On the unconditional positivity of diagonally split Runge-Kutta methods, (Elaydi, S.; Győri, I.; Ladas, G., Advances in Difference Equations (1997), Gordon and Breach), 311-320 · Zbl 0892.65047
[9]	Z. Horváth, On the positivity, contractivity and convergence of diagonally split Runge-Kutta methods on linear problems, Report of Department of Mathematics, Széchenyi István College, Győr, in preparation.; Z. Horváth, On the positivity, contractivity and convergence of diagonally split Runge-Kutta methods on linear problems, Report of Department of Mathematics, Széchenyi István College, Győr, in preparation.
[10]	Hundsdorfer, W., Numerical solution of advection-diffusion-reaction equations, (Lecture notes for Ph.D. course (1996), Thomas Stieltjes Institute), Note NM-N9603 CWI, Amsterdam · Zbl 1030.65100
[11]	Hundsdorfer, W.; Koren, B.; van Loon, M.; Verwer, J., A positive finite-difference advection scheme, J. Comput. Phys., 117, 35-46 (1995) · Zbl 0860.65073
[12]	Hundsdorfer, W.; Spijker, M. N., On the algebraic equations in implicit Runge-Kutta methods, SIAM J. Numer. Anal., 24, 583-594 (1987) · Zbl 0636.65066
[13]	in ’t Hout, K. J., A note on unconditional maximum norm contractivity of diagonally split Runge-Kutta methods, SIAM J. Numer. Anal., 33, 1125-1134 (1996) · Zbl 0855.65095
[14]	Kraaijevanger, J. F.B. M., Contractivity of Runge-Kutta methods, BIT, 31, 482-528 (1991) · Zbl 0763.65059
[15]	Lancaster, P., Theory of Matrices (1969), Academic Press: Academic Press New York · Zbl 0186.05301
[16]	Shampine, L. F., Numerical Solution of Ordinary Differential Equations (1994), Chapman & Hall: Chapman & Hall New York · Zbl 0826.65082
[17]	Shu, C. W.; Osher, S., Efficient implementation of essentially non-oscillatory shock capturing schemes, J. Comput. Phys., 77, 439-471 (1988) · Zbl 0653.65072
[18]	Spijker, M. N., Contractivity in the numerical solution of initial value problems, Numer. Math., 42, 271-290 (1983) · Zbl 0504.65030
[19]	G. Stoyan, Cs. Mihálykó and Zs. Ulbert, Convergence and nonnegativity of numerical methods for an integrodifferential equation describing batch grinding, Comput. Math. Appl., to appear.; G. Stoyan, Cs. Mihálykó and Zs. Ulbert, Convergence and nonnegativity of numerical methods for an integrodifferential equation describing batch grinding, Comput. Math. Appl., to appear.
[20]	Verwer, J., Gauss-Seidel iteration for stiff ODEs from chemical kinetics, SIAM J. Sci. Comput., 15, 1243-1250 (1994) · Zbl 0804.65068

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.