An [formula omitted]-version of the discontinuous Galerkin time-stepping method for nonlinear second-order delay differential equations with vanishing delays

Abstract

We present and analyze an hp-version of the discontinuous Galerkin (DG) time-stepping method for nonlinear second-order delay differential equations with vanishing delays. We derive a priori error bounds in the L2- and L∞-norm that are fully explicit with respect to the local time steps, the local approximation orders, and the local regularity of the exact solution. We further prove that the hp-DG scheme based on geometrically refined time steps and on linearly increasing approximation orders attains exponential rates of convergence for analytic solutions with start-up singularities. Moreover, we also propose an hp-DG scheme for the state dependent delay differential equations. We illustrate the theoretical results with a series of numerical experiments.