Adjoint-based A Posteriori Error Analysis for Semi-explicit Index-1 and Hessenberg Index-2 Differential-Algebraic Equations

TL;DR

This paper develops adjoint-based methods for  posteriori error estimation in the temporal discretization of specific types of differential-algebraic equations, enabling accurate error quantification for quantities of interest.

Contribution

It introduces novel adjoint-based error analysis techniques tailored for semi-explicit index-1 and Hessenberg index-2 DAEs, including conversion to ODEs and direct adjoint formulations.

Findings

High accuracy in error estimation demonstrated through numerical examples

Applicable to nonlinear, non-autonomous DAEs and PDE-DAEs

Two analysis approaches provide flexible error quantification methods

Abstract

In this work we develop adjoint-based analyses for \textit{a posteriori} error estimation for the temporal discretization of differential-algebraic equations (DAEs) of special type: semi-explicit index-1 and Hessenberg index-2. Our technique quantifies the error in a Quantity of Interest (QoI), which is defined as a bounded linear functional of the solution of a DAE. We derive representations for errors of various types of QoIs (depending on the entire time interval, final time, algebraic variables, differential variables, etc.). We develop two analyses: one that defines the adjoint to the DAE system, and one that first converts the DAE to an ODE system and then applies classical \textit{a posteriori} analysis techniques. A number of examples are presented, including nonlinear and non-autonomous DAEs, as well as spatially discretized partial differential-algebraic equations (PDAEs). Numerical results indicate a high degree of accuracy in the error estimation.