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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.

MSC:
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
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