A Superconvergent discontinuous Galerkin method for Volterra integro-differential equations, smooth and non-smooth kernels

TL;DR

This paper develops a superconvergent discontinuous Galerkin method for solving Volterra integro-differential equations with both smooth and non-smooth kernels, providing explicit error bounds and demonstrating superconvergence with graded meshes.

Contribution

It introduces a novel DG method with explicit error bounds for Volterra equations, achieving superconvergence even with non-smooth kernels using graded meshes.

Findings

Error bounds are explicit and depend on problem parameters.

Superconvergence is achieved with graded meshes for non-smooth kernels.

Numerical validation confirms theoretical predictions.

Abstract

We study the numerical solution for Volerra integro-differential equations with smooth and non-smooth kernels. We use a $h$-version discontinuous Galerkin (DG) method and derive nodal error bounds that are explicit in the parameters of interest. In the case of non-smooth kernel, it is justified that the start-up singularities can be resolved at superconvergence rates by using non-uniformly graded meshes. Our theoretical results are numerically validated in a sample of test problems.