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arXiv · 2507.03712

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

Jehanzeb H. Chaudhry·Owen L. Lewis·Md Al Amin Molla
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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.

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Jehanzeb H. Chaudhry, Owen L. Lewis, Md Al Amin Molla. 2025-07-04. Adjoint-based A Posteriori Error Analysis for Semi-explicit Index-1 and Hessenberg Index-2 Differential-Algebraic Equations. https://arxiv.org/abs/2507.03712

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