Nonlinear stability of explicit and diagonally implicit Runge-Kutta methods for neutral delay differential equations in Banach space (Q928100)

The authors investigate the stability and conditional contractivity of explicit and diagonally implicit Runge-Kutta methods for non-linear NDDEs with variable delay in Banach space. They follow the approach designed by Shoufa Li for ODEs, citing the numerical treatment of the ``neutral term'' as the main obstacle in the development of the proof, and then considering two approaches to numerically treating this term. Problem classes \(D(\alpha, \beta, \gamma, L, \lambda^*)\) and \(D_{\delta}(\alpha, \beta, \gamma, L, \lambda^*)\) are introduced in Section 2 and existing results, relevant to the paper, are presented. Section 3 focuses on explicit and diagonally implicit Runge-Kutta methods. Definitions of \(O\)-consistent (for a matrix B and an explicit and diagonally implicit Runge-Kutta method) and \(\beta_0\)-stability are presented. New results, relating to the stability analysis of Runge-Kutta methods applied to \(D(\alpha, \beta, \gamma, L, \lambda^*)\) are presented and proved in Section 4. As a consequence of the arguments being essentially different, compared with the cases of ODEs and DDEs, some numerical results on stiff ODEs with \(\alpha >0\) and stiff DDEs with \(\alpha >0\) are obtained. The stability of the presented methods for problems in class \(D_{\delta}(\alpha, \beta, \gamma, L, \lambda^*)\) is discussed in Section 5. In Section 6 the authors present some illustrative examples, including numerical experiments, which support the results of the paper.