Implicit Runge-Kutta-Nyström Methods with Lagrange Interpolation for Nonlinear Second-Order IVPs with Time-Variable Delay

Abstract

This paper deals with nonlinear second-order initial value problems with time-variable delay. For solving this kind of problems, a class of implicit Runge-Kutta-Nyström (IRKN) methods with Lagrange interpolation are suggested. Under the suitable condition, it is proved that an IRKN method is globally stable and has the computational accuracy $\mathcal{O}(h^{min\{p,\mu+ν+1\}}),$ where $p$ is the consistency order of the method and $\mu, ν ≥0$ are the interpolation parameters. Combining a fourth-order compact difference scheme with IRKN methods, an initial-boundary value problem of nonlinear delay wave equations is solved. The presented experiments further confirm the computational effectiveness of the methods and the theoretical results derived in previous.