A reliable direct numerical treatment of differential–algebraic equations by overdetermined collocation: An operator approach

• Published in: Journal of computational and applied mathematics 2021-05, Vol.387, p.112520, Article 112520
• Main Authors: Hanke, Michael, März, Roswitha
• Format: Article
• Language: English
• Subjects: Algebraic equations; Differential equations; Differential–algebraic operator; Essentially ill-posed problem; Higher index; Higher-order differential–algebraic equation; Ill posed problem; Least squares problems; Least-squares problem; Numerical methods; Overdetermined polynomial collocation; Polynomial collocation; Polynomials
• ISSN: 0377-0427; 1879-1778; 1879-1778
• DOI: 10.1016/j.cam.2019.112520

Recently reported experiments and theoretical contributions concerning overdetermined polynomial collocation applied to higher-index differential–algebraic equations give rise to the conjecture that next to the existing derivative-array based methods there is further potential toward a reliable direct numerical treatment of DAEs. By analyzing first-order differential–algebraic operators and their special approximations in detail, we contribute to justify the overdetermined polynomial collocation applied to first-order higher-index differential–algebraic equations and fill the hitherto existing gap between the theoretical convergence results and its practical realization. Moreover, we shortly touch related questions for higher-order DAEs. We discuss several practical aspects of higher-order differential–algebraic operators and the associated equations which may be important for the application of collocation methods.