Analytical and numerical solutions to higher index linear variable coefficient DAE systems
type: Item
instance of: scholarly article
title: Analytical and numerical solutions to higher index linear variable coefficient DAE systems
zbMATH Open document ID: 0715.65052
DOI: 10.1016/0377-0427(90)90033-V
author: Anders Barrlund
author: Bo Kågström
published in: Journal of Computational and Applied Mathematics
publication date: 1990-01-01T00:00:00Z
review text: The authors investigate the applicability of backward difference algorithms to approximate the solution of singular first order linear systems of ordinary differential equations with variable coefficients. The system considered has the form \((i)\quad A(t)x'(t)+B(t)x(t)=f(t),\) \(x(0)=x_ 0\), \(x(t)\in R^ n\), \(t\geq 0\), det A(t)\(=0\). It is assumed that (i) may be put into the canonical form \((iia)\quad y_ 1'(t)+C(t)y_ 1(t)=f_ 1(t),\) \((iib)\quad N(t)y_ 2'(t)+y_ 2(t)=f_ 2(t),\) \((y_ 1(0),y_ 2(0))\) given, where N(t) is \(s\times s\) nilpotent upper triangular. When the k-step constant stepsize h backward difference algorithm BDF or modified backward difference algorithm MBDF [cf. \textit{L. R. Petzold}, SIAM J. Numer. Anal. 23, 837-852 (1986; Zbl 0635.65084); \textit{K. D. Clark}, SIAM J. Algebraic Discrete Methods 7, 236-246 (1986; Zbl 0594.34003)] are applied to (iia) the result is known to be \(O(h^ k)\) accurate. It is proved here that the same BDF and MBDF methods applied to the \(s\times s\) system (iib), \(k<7\), are also \(O(h^ k)\) accurate globally after a maximum of \((s-1)k+1\) steps. In addition the algorithms are shown to be stable. A detailed account is given of numerical results obtained by applying the BDF, MBDF and (i,j) algorithms to two test problems. The numerical results, which employ PRO-MATLAB software, confirm the theoretical analysis.
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 34A30
zbMATH DE Number: 4178666
zbMATH Keywords: higher index
zbMATH Keywords: DAE systems
zbMATH Keywords: differential-algebraic equations
zbMATH Keywords: backward difference algorithms
zbMATH Keywords: singular first order linear systems
zbMATH Keywords: variable coefficients
zbMATH Keywords: test problems
zbMATH Keywords: numerical results
reviewed by: John B. Butler jun.
describes a project that uses: Matlab
describes a project that uses: DASSL
MaRDI profile type: Publication
cites work: Q3940107
cites work: One canonical form for higher-index linear time-varying singular systems
cites work: Index Two Linear Time-Varying Singular Systems of Differential Equations
cites work: The Numerical Solution of Higher Index Linear Time Varying Singular Systems of Differential Equations
cites work: Canonical Forms and Solvable Singular Systems of Differential Equations
cites work: Difference Methods for the Numerical Solution of Time-Varying Singular Systems of Differential Equations
cites work: Q3742613
cites work: Computing stable eigendecompositions of matrix pencils
cites work: Q3254327
cites work: The automatic integration of ordinary differential equations
cites work: Q3957868
cites work: ODE Methods for the Solution of Differential/Algebraic Systems
cites work: Bounds and perturbation bounds for the matrix exponential
cites work: Dynamic equations in descriptor form
cites work: Q3950823
cites work: Differential/Algebraic Equations are not ODE’<scp>s</scp>
cites work: Order Results for Implicit Runge–Kutta Methods Applied to Differential/Algebraic Systems
cites work: Analysis of descriptor systems using numerical algorithms
cites work: Stability and Asymptotic Estimates in Nonautonomous Linear Differential Systems
cites work: On Logarithmic Norms
cites work: The Sensitivity of the Matrix Exponential
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description: scientific article; zbMATH DE number 4178666