An $hp$-Version of $C^0$ -Continuous Petrov-Galerkin Time-Stepping Method for Second-Order Volterra Integro-Differential Equations with Weakly Singular Kernels

Abstract

An $hp$-version of $C^0$-CPG time-stepping method for second-order Volterra integro-differential equations with weakly singular kernels is studied. In contrast to the methods reducing second-order problems to first-order systems, here the CG and DG methodologies are combined to directly discretise the second-order derivative. An a priori error estimate in the $H^1$-norm, fully explicit with respect to the local discretisation and regularity parameters, is derived. It is shown that for analytic solutions with start-up singularities, exponential rates of convergence can be achieved by using geometrically refined time steps and linearly increasing approximation orders. Theoretical results are illustrated by numerical examples.