On the positivity step size threshold of Runge--Kutta methods
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.
scientific article; zbMATH DE number 2160267