Special boundedness properties in numerical initial value problems
This paper is concerned with the analysis of some nonlinear stability properties of discretization methods for the numerical solution of initial value problems (IVPs) for differential systems. The proposed theory allows the authors to deal with standard classes of discretization methods for IVPs such as linear multistep and Runge-Kutta methods in connection with many stability properties studied along the last three decades for different classes of problems such as total-variation diminishing, strong stability preservation and boundedness. The authors consider general IVPs in a vector space \(\mathbf{V}\) with the form (1) \(\dot{u} (t) = F(t, u(t)),\) \( t \geq 0\), \( u(0)= u_0 \in \mathbf{V}\), and discretization methods that can be written as \((2) y_i = \sum_{j=1}^l \sigma_{ij} x_j + \Delta t \; \sum_{j=1}^m \tau_{ij} F( \overline{t}_j, y_j)\), \( i=1, \dots ,m\), where \( x_j \in \mathbf{V}\), \(j=1, \dots ,l\) are the input vectors with the information on the solution in the current step, \( \Delta t\) the step size, \( y_j \simeq u ( \overline{t}_j)\), \(j=1, \dots ,m\) are approximations to the solution at the time levels \( \overline{t}_j\) and \( S = ( \sigma_{ij})\), \( T = ( \tau_{ij})\) real matrices that define the method. As remarked most standard methods can be written in the form (2) with suitable \(S\) and \(T\) matrices. In this setting, assuming that for given \( \tau_0 >0\), the inequality \( \| v + \tau_0 \; F( \overline{t}_j, v) \| \leq \| v \| \) holds for \( j=1, \dots, m\) and and all \( v \in \mathbf{V}\) the authors derive conditions on the step size guaranteeing bounds on the solution of (2) of type \( \| y_i \| \leq \mu_i \max_{j=1, \dots ,l} \| x_j \|\), \((i=1, \dots ,m)\) and \( \| y_i \| \leq \sum_{j=1}^l \mu_{ij} \| x_j \|\), \((i=1, \dots ,m)\) for non negative coefficients \( \mu_i\) and \( \mu_{ij}\) where the functional \( \| \cdot \| \) can be a seminorm or even more general sublinear functionals. Among the most relevant conclusions of the theoretical stability analysis presented here we may mention the following ones: 1) The theory provides a unified framework to deal with a variety of monotonicity results given in the literature for seminorms or convex functionals. 2) In addition this theory goes further by considering also sublinear functionals. 3) Optimal bounds on the step size have been derived at least in general functionals that can be checked more easily. 4) Some proofs of earlier monotonicity results turn out to be surprisingly simple. 5) Finally, the remarks on section 4 on the two step Adams method provide a deep view of many aspects of this theory.
scientific article; zbMATH DE number 5996280
scientific article