zbMATH — the first resource for mathematics

On the positivity step size threshold of Runge–Kutta methods. (English) Zbl 1073.65077
The author considers the initial value problem (IVP) \[ U'(t)- f(t,U(t)),\quad t\geq t_0,\quad U(t_0)= u_0, \] where \(t_0\in\mathbb{R}\), \(u_0\in\mathbb{R}^n\), \(f:\mathbb{R}\times \mathbb{R}^n\to \mathbb{R}^n\).
IVP is called positive if \(U(t)\geq 0\) for \(t\in [t_0, t^*]\) and \(f\in P\). For the numerical solution of the IVP the general Runge-Kutta methods (RK) with \(s\) stages are used, defined by a matrix \(A= (a_{ij})\) and vectors \(b= (b_i)\), \(c= (c_i)\), \(i,j= 1,2,\dots, s\).
\(H\) is called the step size threshold of positivity, if for step \(h_{im}\leq H\) the solutions \(u_m\) obtained by this method, are positive for any \(m\) finite from \(\mathbb{R}\).
For an irreducible non-confluent RK-method the threshold \(H\) is equivalent with the positivity radius \(R(A,b)\), introduced and studied thoroughly in numerical analysis literature.
In the second part the positive invariant cone for a linear problem and for a quasilinear problem is constructed.

65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
34C14 Symmetries, invariants of ordinary differential equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
Full Text: DOI
[1] Bermann, A.; Plemmons, R.J., Nonnegative matrices in the mathematical sciences, (1979), Academic Press New York
[2] Berzins, M., Modified mass matrices and positivity preservation for hyperbolic and parabolic pdes, Commun. numer. methods engrg., 17, 9, 659-666, (2001) · Zbl 0986.65091
[3] Bolley, C.; Crouzeix, M., Conservation de la positivité lors de la discrétisation des problèmes d’évolution paraboliques, RAIRO anal. numér., 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 Chichester · Zbl 0616.65072
[5] Chawla, M.M.; Al-Zanaidi, M.A.; Evans, D.J., Generalized trapezoidal formulas for parabolic equations, Internat. J. comput. math., 70, 3, 429-443, (1999) · Zbl 0926.65081
[6] Dekker, K.; Kraaijevanger, J.F.B.M.; Schneid, J., On the relation between algebraic stability and B-convergence for runge – kutta methods, Numer. math., 57, 249-262, (1990) · Zbl 0702.65079
[7] Faragó, I.; Horváth, R.; Korotov, S., Discrete maximum principle for linear parabolic problems solved on hybrid meshes, Appl. numer. math., 53, 2-4, 249-264, (2005), (this issue) · Zbl 1070.65094
[8] L. Ferracina, M.N. Spijker, Stepsize restrictions for the total-variation-diminishing property in general Runge-Kutta methods, Report MI 2002-21, University of Leiden · Zbl 1080.65087
[9] Gerisch, A.; Griffiths, D.F.; Weiner, R.; Chaplain, M.A.J., A positive splitting method for mixed hyperbolic – parabolic systems, Numer. methods partial differential equations, 17, 2, 152-168, (2001) · Zbl 0981.65112
[10] Gerisch, A.; Weiner, R., The positivity of low order explicit runge – kutta schemes applied in splitting methods, Comput. math. appl., 45, 53-67, (2003) · Zbl 1036.65061
[11] Hairer, E.; Wanner, G., Solving ordinary differential equations II, (1991), Springer Berlin · Zbl 0729.65051
[12] Horváth, Z., Positivity of runge – kutta and diagonally split runge – kutta methods, Appl. numer. math., 28, 309-326, (1998) · Zbl 0926.65073
[13] Horváth, Z., On the positivity of matrix – vector products, Linear algebra appl., 393, 253-258, (2004) · Zbl 1070.15018
[14] Hundsdorfer, W., Numerical solution of advection – diffusion – reaction equations, Lecture notes for ph.D. course, Thomas Stieltjes institute, note NM-N9603, (1996), CWI Amsterdam
[15] Hundsdorfer, W.; Verwer, J., Numerical solution of time-dependent advection – diffusion – reaction equations, Springer series in computational mathematics, vol. 33, (2003), Springer Berlin · Zbl 1030.65100
[16] Kraaijevanger, J.F.B.M., Contractivity of runge – kutta methods, Bit, 31, 482-528, (1991) · Zbl 0763.65059
[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] Smith, H.L., Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems, (1995), American Mathematical Society Providence, RI · Zbl 0821.34003
[19] Stoyan, G.; Mihálykó, C.; Ulbert, Z., Convergence and nonnegativity of numerical methods for an integrodifferential equation describing batch grinding, Comput. math. appl., 35, 12, 69-81, (1998) · Zbl 0999.65151
[20] Stuart, A., Perturbation theory for infinite dimensional dynamical systems, (), 181-290 · Zbl 0847.34070
[21] Vejchodský, T., On the nonnegativity conservation in semidiscrete parabolic problems, () · Zbl 1073.65096
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.