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 guaran- teed bounds for the uncertainties are known, interval meth- ods 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 arith- metic, as well as simulation techniques employing split- ting 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 re- sults for simplified system models in mechanical and bio- process engineering show specific properties, advantages, and limitations of each tool.