On the positivity step size threshold of Runge–Kutta methods

Horváth, Zoltán

doi:10.1016/j.apnum.2004.08.026

Cited by 24 publications

(28 citation statements)

References 12 publications

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“…In this Subsection, we propose a method for constructing positive ENRK methods based on the results of positivity of Runge-Kutta methods [18,19].…”

Section: Positive Enrk Methodsmentioning

confidence: 99%

“…Besides, ENRK4 and (based on the classical 4-stages RK method) are also considered although these methods do not possess R(A, b) > 0. It is easy to verify that the right-hand side f of the model belongs to the set P α := {f |f (t, v) + αv ≥ 0 for all t, v ≥ 0} with α = max{A − 1, D} = 1 (see [18,19]), therefore it is easy to determine positivity step size thresholds for ENRK methods.…”

Section: A Predator-prey System With Beddington-deangelis Functional mentioning

confidence: 99%

“…In this paper, first a PES step size threshold for ESRK methods are determined on the base of the stability analysis of ESRK methods and the application of results for positive Runge-Kutta methods [18,19]. After that the choice of nonstandard denominator functions for preserving the accuracy order of the original ESRK methods will be investigated.…”

Section: Introductionmentioning

confidence: 99%

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Positive and elementary stable explicit nonstandard Runge-Kutta methods for a class of autonomous dynamical systems

Dang¹,

Hoang²

2019

International Journal of Computer Mathematics

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In this paper, we construct explicit nonstandard Runge-Kutta (ENRK) methods which have higher accuracy order and preserve two important properties of autonomous dynamical systems, namely, the positivity and linear stability. These methods are based on the classical explicit Runge-Kutta methods, where instead of the usual h in the formulas there stands a function ϕ(h). It is proved that the constructed methods preserve the accuracy order of the original Runge-Kutta methods. The numerical simulations confirm the validity of the obtained theoretical results. difference schemes; Positive nonstandard finite difference methods; Elementary stable. Our first objective is to construct difference schemes preserving the linear stability of the equilibrium points of System (1) for all finite step-size h > 0. These schemes are called also elementary stable [3,7,8]. It should be emphasized that standard finite difference schemes cannot preserve properties of the differential equations for any step-sizes h > 0, including the linear stability. Mickens called this phenomenon numerical instability [25].The construction of elementary stable difference schemes play especially important role in numerical solution of differential equations and numerical simulation of nonlinear dynamical systems. In 2005, Dimitrov and Kojouharov [7] proposed a method for constructing elementary stable NSFD methods for general two-dimensional autonomous dynamical systems. These NSFD methods are based on the explicit and implicit Euler and the second-order Runge-Kutta methods. Later, in 2007 these results are extented for the general n-dimensional dynamical systems, namely, NSFD schemes preserving elementary stability are constructed based on the θ-methods and the second-order Runge-Kutta methods [8]. One important action in the construction of the elementary stable NSFD schemes is the replacement of the standard denominator function ϕ(h) = h by the nonstandard denominator function, which is Email addresses: dangquanga@cic.vast.vn (Quang A Dang),

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“…In this Subsection, we propose a method for constructing positive ENRK methods based on the results of positivity of Runge-Kutta methods [18,19].…”

Section: Positive Enrk Methodsmentioning

confidence: 99%

Section: A Predator-prey System With Beddington-deangelis Functional mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Positive and elementary stable explicit nonstandard Runge-Kutta methods for a class of autonomous dynamical systems

Dang¹,

Hoang²

2019

International Journal of Computer Mathematics

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“…when C = [0, ∞) N , for a survey see e.g. [4,3] and it seems no general theory exists yet concerning Ros-type methods. 2) and solving formally to y i we obtain for n = 0…”

Section: Rosenbrock-type Methodsmentioning

confidence: 99%

Positively invariant cones of dynamical systems under Runge‐Kutta and Rosenbrock‐type discretization

Horváth

2004

Proc Appl Math and Mech

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In this paper we consider positively invariant cones of finite dimensional dynamical systems and study conditions on the time step-size that guarantee the discrete positive invariance of these cones under Runge-Kutta and Rosenbrock-type methods. We conclude quite simple sufficient conditions, which involve the positivity (or absolute monotonicity) radius of the Runge-Kutta schemes and its generalization when the Rosenbrock-type methods are applied.

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“…For unconstrained systems, i.e., for systems where E equals the identity map I ∈ R n×n , this topic is already well studied, see eg. [3,2] and the references therein. For systems where additional conditions like balance equations or conservation laws lead to a differential-algebraic structure, this topic is new.…”

Section: Introductionmentioning

confidence: 99%

Positivity inheritance for linear problems

Baum

Mehrmann

2010

Proc Appl Math and Mech

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We discuss the numerical solution of positive Differential-Algebraic-Equations (DAEs). For Ordinary Differential Equations (ODEs) where the system matrix is a -M-Matrix, Runge-Kutta-or Multistep-Method are positive if the stepsize is chosen within the absolutely monotonicity radius of the considered method. We extend this concept to matrix pairs and present conditions for positivity preserving discretizations of linear, time-invariant DAEs.

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On the positivity step size threshold of Runge–Kutta methods

Cited by 24 publications

References 12 publications

Positive and elementary stable explicit nonstandard Runge-Kutta methods for a class of autonomous dynamical systems

Positive and elementary stable explicit nonstandard Runge-Kutta methods for a class of autonomous dynamical systems

Positively invariant cones of dynamical systems under Runge‐Kutta and Rosenbrock‐type discretization

Positivity inheritance for linear problems

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