Strang-type preconditioners applied to ordinary and neutral differential-algebraic equations. (English) Zbl 1249.65165

Zhang, Chengjian; Chen, Hao; Wang, Leiming

Numer. Linear Algebra Appl. 18, No. 5, 843-855 (2011).

The authors apply the theory on boundary value methods (BVMs) to numerically solve linear ordinary and neutral differential-algebraic equations.
The coefficient matrix of the linear systems obtained after the discretization procedure has the structure of a sparse block quasi-Toeplitz matrix. From the literature, it is known that the solution of such kind of systems may be conveniently carried out by the aid of Strang preconditioners or variants of them, which accelerate the convergence rate of iterative methods such as the GMRES method.
The authors borrow these techniques and show, from both a theoretical and numerical viewpoint, their effectiveness when applied to these classes of continuous problems. In particular they give conditions on both the method and the continuous problem which assure the invertibility of the preconditioners as well as a significant clustering of the spectra of the corresponding preconditioned systems. A few examples are reported to confirm the theoretical results.

Reviewer: Felice Iavernaro (Bari)

MSC:

65L80	Numerical methods for differential-algebraic equations
65F08	Preconditioners for iterative methods
34A30	Linear ordinary differential equations and systems
34A09	Implicit ordinary differential equations, differential-algebraic equations
34K40	Neutral functional-differential equations
34K28	Numerical approximation of solutions of functional-differential equations (MSC2010)
65L03	Numerical methods for functional-differential equations
65L20	Stability and convergence of numerical methods for ordinary differential equations

Keywords:

neutral differential-algebraic equation; linear system; Strang-type preconditioner; boundary-value method; convergence rate; numerical examples; sparse block quasi-Toeplitz matrix; iterative methods; GMRES method

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