Impasse Points

Overview

Impasse Points is a Python implementation of the Rabier & Rheinboldt (1994) algorithm for integrating semi-explicit differential-algebraic equations (DAEs) through singularities.

$$\begin{aligned} A_1(x) \dot{x} &= G_1(x) \\ G_2(x) &= 0 \end{aligned}$$

An impasse point is a singularity at which the algebraic constraints prevent the solution from continuing beyond a certain time $T^*$, even though the solution remains bounded. These singularities often indicate that the model is idealized. Standard solvers fail at impasse points. This algorithm detects them by monitoring a scalar $\gamma$ that vanishes when the system becomes singular, and can traverse the singularity to explore other solution branches using a pseudo-time reparametrization.

Key Features

Singularity Traversal

The algorithm enables integration through impasse points using:

• Pseudo-time parametrization: Replaces physical time $t$ with a pseudo-time $s$ where $dt/ds = \gamma(x)$. As the system approaches an impasse point, $\gamma \to 0$ but the solution in pseudo-time remains smooth.
• Augmented system formulation: Adds normalization conditions to handle rank-deficient matrices, inspired by numerical bifurcation theory.
• Gamma sign monitoring: Detects impasse points by tracking sign changes in $\gamma$, which vanishes at singularities.

Numerical Methods

• Constraint projection: Newton iteration to maintain solutions on the constraint manifold
• QR factorization: Computes orthonormal bases for tangent and normal spaces
• Adaptive integration: DOPRI5 Runge-Kutta with error control

Reference

P.J. Rabier and W.C. Rheinboldt, On the Computation of Impasse Points of Quasi-Linear Differential-Algebraic Equations, Mathematics of Computation, Vol. 62, No. 205, pp. 133-154, January 1994.