The numerical solution of some higher-index time-varying semistate systems by difference methods
Backwards differentiation formulas (BDF) are one of the more popular methods used on singular systems of differential equations (also known as singular or descriptor systems). It known that even in the linear time varying case that BDF approximations need not converge. This paper examines the behavior of BDF on linear time varying semi-state systems of the form \(A(t)x'(t)+B(t)x(t)=f(t).\) Necessary and sufficient conditions are given for the solvability of fully implicit index one and semi- explicit index two linear systems in this form. Using this solvability criteria, the convergence and stability of BDF for solvable semi-explicit index two systems is established. These methods exhibit a boundary layer of instability of length 2k (where k is the order of the method) in the algebraic variables. Finally, it is shown that these systems belong to a class of BDF solvable systems which strictly contains the systems safely transformable to standard canonical form.
fully implicit index one and semi-explicit index two linear systems
scientific article; zbMATH DE number 4018259
scientific article