The Hp Time Step Method For The Time-delayed Volterra Integro-differential Equation

In this paper,we study the h-p version time stepping methods for linear Volterra integro-differential equations with delays.Firstly,we propose an h-p version of the continuous Petrov-Galerkin method for the linear Volterra integro-differential equations with proportional delays.We derive a-priori error estimates in the L2-,H1-and L?-norms.that are explicit in the local time steps,the local polynomial degrees,and the local regularity of the exact solution.Secondly,we develop an h-p version of the discontinuous Galerkin method for the linear Volterra integro-differential equations with vanishing delays.We also obtain a-priori error estimates in the L2-and L00-norms that are completely explicit with respect to the local discretization and regularity parameters.In particular,we prove that the h-p version discontinuous Galerkin scheme based on geometrically refined time steps and on linearly increasing approximation orders can achieve exponential rates of convergence for solutions with start-up singularities.Finally,we introduce an h-p version of the Chebyshev-Gauss-Lobatto spectral collocation method for the linear Volterra integro-differential equations with vanishing delays.We design a fast algorithm with high accuracy and derive an h-p version error estimate in the H1-norm.Several numerical experiments are carried out to confirm the above theoretical results.