VALENCIA-IVP: A Comparison with Other Initial Value Problem Solvers

Validated integration of ordinary differential equations with uncertain initial conditions and uncertain parameters is important for many practical applications. If guaranteed bounds for the uncertainties are known, interval methods can be applied to obtain validated enclosures of all states. However, validated computations are often affected by overestimation, which, in naive implementations, might even lead to meaningless results. Parallelepiped and QR preconditioning of the state equations, Taylor model arithmetic, as well as simulation techniques employing splitting and merging routines are a few existing approaches for reduction of overestimation. In this paper, the recently developed validated solver ValEncIA-IVP and several methods implemented there for reduction of overestimation are described. Furthermore, a detailed comparison of this solver with COSY VI and VNODE, two of the most well- known validated ODE solvers, is presented. Simulation results for simplified system models in mechanical and bio- process engineering show specific properties, advantages, and limitations of each tool.