Analytical and numerical solutions to higher index linear variable coefficient DAE systems
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.
scientific article; zbMATH DE number 4178666
scientific article