On the convergence of iterative methods for general differential-algebraic systems
The main object of the paper is the system of functional-integral-algebraic equations \[ \dot {x}(t) = x^0 + \int_{0}^{t} f(t,s,x(\cdot),\lambda(\cdot))\,ds, \quad t\in J=[0,T], \tag{1} \] \[ \lambda(t)=g(t,x(\cdot),\lambda(\cdot)), \quad t\in J, \tag{2} \] where all functions and functionals are continuous (in spaces with the maximum norms). The purpose of the paper is the investigation of conditions that provide existence and uniqueness of a solution to (1), (2) as well as convergence of different iterative procedures to it. The most attention is devoted to the waveform relaxation (WR) algorithm for resolving nonlinear equations, that is based on the transition from the initial nonlinear functions of the system to be solved to equivalent splitting ones via their variables set expansion. This paper is developing results obtained earlier by two of the authors for differential-algebraic equations (DAEs), which are defined by operators of Volterra type [cf. \textit{Z. Jackiewicz} and \textit{M. Kwapisz}, SIAM J. Numer. Anal. 33, No. 6, 2303--2317 (1996; Zbl 0889.34064)]. Here this property is omitted. Besides the WR the classical Picard and Gauss-Seidel algorithms are considered. Special cases of problems (1), (2), particularly a quasi-linear DAEs are also investigated and numerical results are presented.
scientific article; zbMATH DE number 2091417