Validated simulation of differential algebraic equations with runge-kutta methods

Differential Algebraic Equations (DAEs) are a general and implicit form of differential equations. DAEs are often used to represent physical systems such as dynamics of solids or chemical interactions. These equations are different from Ordinary Differential Equations (ODEs) in the sense that some of the dependent variables occur without their derivatives. Validated simulation of ODEs has recently been the subject of different developments such as guaranteed Runge-Kutta integration schemes, both explicit and implicit. Not so different from solving an ODE, solving a DAE consists of searching for a consistent initial value and computing a trajectory. Nevertheless, DAEs are in generally much more difficult to solve than ODEs. In this paper, we focus on the semi-explicit form of index one, called Hessenberg index-1 form. We propose a validated way to simulate this kind of differential equations. Finally, our method is applied to different examples in order to show its efficiency.