Stability of Discontinuous Galerkin Methods for Volterra Integral Equations

We conduct the stability analysis of discontinuous Galerkin methods applied to Volterra integral equations in this paper. Stability conditions with respect to both the basic and convolution test equations are derived. Our findings indicate that the methods with orders up to 6 exhibit A$$ A $$-stability when applied with the basic test equation, while demonstrating unbounded stability regions when applied to the convolution test equation. Additionally, the results of V0$$ {V}_0 $$-stability for the semidiscretized variants (quadrature discontinuous Galerkin methods) and fully discretized versions (fully discretized discontinuous Galerkin methods) with orders 1 and 2 are presented when solving the convolution test equation. To corroborate these theoretical results, we provide some numerical experiments for validation.