An approximation for nonlinear differential-algebraic equations via singular perturbation theory

TL;DR

This paper introduces a novel approximation method for nonlinear differential-algebraic equations (DAEs) using singular perturbation theory, providing a coordinate-invariant approach to handle inconsistent initial conditions and jumps.

Contribution

It proposes a new nonlinear Weierstrass form and a generalized nonlinear consistency projector, along with a singular perturbation approximation that effectively captures solution jumps and approximates DAE solutions.

Findings

The nonlinear consistency projector is coordinate-invariant.

The singular perturbation system approximates jumps and DAE solutions.

Numerical simulation confirms the method's effectiveness.

Abstract

In this paper, we study jumps of nonlinear DAEs caused by inconsistent initial values. First, we propose a simple normal form called the index-1 nonlinear Weierstrass form (INWF) for nonlinear DAEs. Then we generalize the notion of consistency projector for linear DAEs to the nonlinear case. By an example, we compare our proposed nonlinear consistency projectors with two existing consistent initialization methods to show that the two existing methods are not coordinate-free, i.e., the consistent points calculated by the two methods are not invariant under nonlinear coordinates transformations. Next we propose a singular perturbed system approximation for nonlinear DAEs, which is an ordinary differential equation (ODE) with a small perturbation parameter, we show that the solutions of the proposed perturbation system approximate both the jumps resulting from the nonlinear consistency projectors and the $\mathcal C^1$-solutions of the DAE. At last, we use a numerical simulation of a nonlinear DAE model arising from an electric circuit to illustrate the effectiveness of the proposed singular perturbed system approximation of DAEs.