Least-squares collocation for linear higher-index differential-algebraic equations. (English) Zbl 1357.65106
Summary: Differential-algebraic equations with higher index give rise to essentially ill-posed problems. Therefore, their numerical approximation requires special care. In the present paper, we state the notion of ill-posedness for linear differential-algebraic equations more precisely. Based on this property, we construct a regularization procedure using a least-squares collocation approach by discretizing the pre-image space. Numerical experiments show that the resulting method has excellent convergence properties and is not much more computationally expensive than standard collocation methods used in the numerical solution of ordinary differential equations or index-1 differential-algebraic equations. Convergence is shown for a limited class of linear higher-index differential-algebraic equations.
MSC:
| 65L80 | Numerical methods for differential-algebraic equations |
| 65L60 | Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations |
| 65L08 | Numerical solution of ill-posed problems involving ordinary differential equations |
| 65L05 | Numerical methods for initial value problems involving ordinary differential equations |
| 65L10 | Numerical solution of boundary value problems involving ordinary differential equations |
| 34A09 | Implicit ordinary differential equations, differential-algebraic equations |
| 34A30 | Linear ordinary differential equations and systems |
| 65L20 | Stability and convergence of numerical methods for ordinary differential equations |
Keywords:
differential-algebraic equation; higher index; essentially ill-posed problem; collocation; boundary value problem; initial value problem; numerical experiment; convergence; linearSoftware:
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