scholarly journals Strongly A-stable first stage explicit collocation methods with stepsize control for stiff and differential–algebraic equations

2014 ◽  
Vol 259 ◽  
pp. 138-152 ◽  
Author(s):  
S. González-Pinto ◽  
D. Hernández-Abreu ◽  
B. Simeon
Keyword(s):  
Differential Algebraic Equations ◽  
Collocation Methods ◽  
Algebraic Equations ◽  
Stepsize Control

Collocation methods for integro-differential algebraic equations with index 1

2019 ◽  
Vol 40 (2) ◽  
pp. 850-885 ◽  
Author(s):  
Hui Liang ◽  
Hermann Brunner
Keyword(s):  
Exact Solution ◽  
Differential Algebraic Equations ◽  
Collocation Methods ◽  
Polynomial Space ◽  
Algebraic Equations ◽  
Piecewise Polynomial ◽  
Tractability Index ◽  
Volterra Integral Operator ◽  
The Given ◽  
Collocation Solution

Abstract The notion of the tractability index based on the $\nu $-smoothing property of a Volterra integral operator is introduced for general systems of linear integro-differential algebraic equations (IDAEs). It is used to decouple the given IDAE system of index $1$ into the inherent system of regular Volterra integro-differential equations (VIDEs) and a system of second-kind Volterra integral equations (VIEs). This decoupling of the given general IDAE forms the basis for the convergence analysis of the two classes of piecewise polynomial collocation methods for solving the given index-$1$ IDAE system. The first one employs the same continuous piecewise polynomial space $S_m^{(0)}$ for both the VIDE part and the second-kind VIE part of the decoupled system. In the second one the VIDE part is discretized in $S_m^{(0)}$, but the second-kind VIE part employs the space of discontinuous piecewise polynomials $S_{m - 1}^{(- 1)}$. The optimal orders of convergence of these collocation methods are derived. For the first method, the collocation solution converges uniformly to the exact solution if and only if the collocation parameters satisfy a certain condition. This condition is no longer necessary for the second method; the collocation solution now converges to the exact solution for any choice of the collocation parameters. Numerical examples illustrate the theoretical results.

PERIODIC SOLUTIONS OF DIFFERENTIAL ALGEBRAIC EQUATIONS WITH TIME DELAYS: COMPUTATION AND STABILITY ANALYSIS

2006 ◽  
Vol 16 (01) ◽  
pp. 67-84 ◽  
Author(s):  
TATYANA LUZYANINA ◽  
DIRK ROOSE
Keyword(s):  
Stability Analysis ◽  
Periodic Solutions ◽  
Time Delays ◽  
Differential Algebraic Equations ◽  
Collocation Methods ◽  
Algebraic Equations ◽  
Floquet Multipliers ◽  
Infinite Dimensional ◽  
Local Asymptotic Stability ◽  
Convergence Results

This paper concerns the computation and local stability analysis of periodic solutions to semi-explicit differential algebraic equations with time delays (delay DAEs) of index 1 and index 2. By presenting different formulations of delay DAEs, we motivate our choice of a direct treatment of these equations. Periodic solutions are computed by solving a periodic two-point boundary value problem, which is an infinite-dimensional problem for delay DAEs. We investigate two collocation methods based on piecewise polynomials: collocation at Radau IIA and Gauss–Legendre nodes. Using the obtained collocation equations, we compute an approximation to the Floquet multipliers which determine the local asymptotic stability of a periodic solution. Based on numerical experiments, we present orders of convergence for the computed solutions and Floquet multipliers and compare our results with known theoretical convergence results for initial value problems for delay DAEs. We end with examples on bifurcation analysis of delay DAEs.

scholarly journals Towards a reliable implementation of least-squares collocation for higher index differential-algebraic equations—Part 2: the discrete least-squares problem

2021 ◽  
Author(s):  
Michael Hanke ◽  
Roswitha März
Keyword(s):  
Least Squares ◽  
Differential Algebraic Equations ◽  
Collocation Methods ◽  
Discrete Problem ◽  
Algebraic Equations ◽  
Least Squares Collocation ◽  
Least Squares Problem ◽  
Ill Posed ◽  
Natural Settings ◽  
Selection Of

AbstractIn the two parts of the present note we discuss questions concerning the implementation of overdetermined least-squares collocation methods for higher index differential-algebraic equations (DAEs). Since higher index DAEs lead to ill-posed problems in natural settings, the discrete counterparts are expected to be very sensitive, which attaches particular importance to their implementation. We provide in Part 1 a robust selection of basis functions and collocation points to design the discrete problem whereas we analyze the discrete least-squares problem and substantiate a procedure for its numerical solution in Part 2.

scholarly journals Towards a reliable implementation of least-squares collocation for higher index differential-algebraic equations—Part 1: basics and ansatz function choices

2021 ◽  
Author(s):  
Michael Hanke ◽  
Roswitha März
Keyword(s):  
Least Squares ◽  
Differential Algebraic Equations ◽  
Collocation Methods ◽  
Discrete Problem ◽  
Algebraic Equations ◽  
Least Squares Collocation ◽  
Ill Posed ◽  
Present Part ◽  
Natural Settings ◽  
Selection Of

AbstractIn the two parts of the present note we discuss several questions concerning the implementation of overdetermined least-squares collocation methods for higher index differential-algebraic equations (DAEs). Since higher index DAEs lead to ill-posed problems in natural settings, the discrete counterparts are expected to be very sensitive, which attaches particular importance to their implementation. In the present Part 1, we provide a robust selection of basis functions and collocation points to design the discrete problem. We substantiate a procedure for its numerical solution later in Part 2. Additionally, in Part 1, a number of new error estimates are proven that support some of the design decisions.

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