A Symbolic–Numerical Method for Integration of DAEs Based on Geometric Control Theory

This work describes a symbolic–numerical integration method for a class of differential algebraic equations (DAEs) known as semi-explicit systems. Our method relies on geometric theory of decoupling for nonlinear systems combined with efficient numerical analysis techniques. It uses an algorithm that applies symbolic and numerical calculations to build an explicit vector field \(\tau \), whose integral curves with compatible initial conditions are the same solutions of the original DAE. Here, compatible initial conditions are the ones that respect the algebraic restrictions and their derivatives up to their relative degree. This extended set of restrictions defines a submanifold \(\varGamma \) of the whole space, which is formed by all the variables of the system, and all solutions of the DAE lie on this submanifold. Furthermore, even for nonexactly compatible initial conditions, the solutions of this explicit system defined by \(\tau \) converge exponentially to \(\varGamma \). Under mild assumptions, an approximation result shows that the precision of the method is essentially controlled by the distance of the initial condition from \(\varGamma \). A scheme to compute compatible initial conditions with the DAE is also provided. Finally, simulations with benchmarks and comparisons with other available methods show that this is a suitable alternative for these problems, specially for nonexactly compatible initial conditions or high-index problems.