Continuous Galerkin Schemes for Semi-Explicit Differential-Algebraic Equations

TL;DR

This paper introduces a new class of continuous Galerkin integration schemes for semi-explicit differential-algebraic equations of index 2, emphasizing variational consistency and proven convergence properties.

Contribution

It develops flexible, variationally consistent Galerkin schemes for semi-explicit DAE systems, with proven convergence for linear constraints and numerical verification.

Findings

Convergence of order r+1 for states and multipliers with polynomial degree r.

Schemes allow different discretizations for differential and algebraic parts.

Numerical experiments confirm theoretical convergence results.

Abstract

This paper studies a new class of integration schemes for the numerical solution of semi-explicit differential-algebraic equations of differentiation index 2 in Hessenberg form. Our schemes provide the flexibility to choose different discretizations in the differential and algebraic equations. At the same time, they are designed to have a property called variational consistency, i.e., the choice of the discretization of the constraint determines the discretization of the Lagrange multiplier. For the case of linear constraints, we prove convergence of order r+1 both for the state and the multiplier if piecewise polynomials of order r are used. These results are also verified numerically.