On the blunting method in verified integration of ODEs

Verified methods for the integration of initial value problems (IVPs) for ordinary differential equations (ODEs) aim at computing guaranteed error bounds for the flow of an ODE, while maintaining a low level of overestimation. This paper is concerned with one of the sources of overestimation: a matrix-vector product describing a parallelepiped in phase space. We analyze the blunting method developed by Berz and Makino, which consists of a special choice of the matrix in this product. For the linear model problem u′ = Au, u(0) = u0 ∈ u0, where u ∈ ℝ^m, A ∈ ℝ^{m x m}, m ≥ 2, and u0 is a given interval vector, we compare the convergence behavior of the blunting method with that of the well-known QR method. In both methods, the amount of overestimation of the flow of the initial set depends on the spectral radius of some well-defined matrix. We show that under certain conditions, the spectral radii of the matrices that describe the excess propagation in the QR method and in the blunting method have the same limits, and the excess propagation in both methods is similar.