Validated Simulation of Differential Algebraic Equations with Runge-Kutta Methods

Abstract : Differential Algebraic Equations (DAEs) are a general and implicit form of differential equations. This mathematical object is often used to represent physical systems such as dynamics of solid or chemical interactions. These equations are different from Ordinary Differential Equations (ODEs) in sense that some of the dependent variables occur without their derivatives. These variables are called ``algebraic variables'', which means free of derivatives and not with respect to abstract algebra. Validated simulation of ODEs has recently known different developments such as guaranteed Runge-Kutta integration schemes, explicit and implicit ones. Not so far from an ODE, solving a DAE consists of searching 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.
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Reliable Computing electronic edition, 2016, Special issue devoted to material presented at SWIM 2015, 22, 〈http://interval.louisiana.edu/reliable-computing-journal/volume-22/reliable-computing-22-pp-056-077.pdf〉
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Julien Alexandre Dit Sandretto, Alexandre Chapoutot. Validated Simulation of Differential Algebraic Equations with Runge-Kutta Methods. Reliable Computing electronic edition, 2016, Special issue devoted to material presented at SWIM 2015, 22, 〈http://interval.louisiana.edu/reliable-computing-journal/volume-22/reliable-computing-22-pp-056-077.pdf〉. 〈hal-01243044〉

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