Uniform l1 behavior of the first-order interpolant quadrature scheme for some partial integro-differential equations

Abstract We study the time discretization for the solution of a Volterra equation with weakly singular kernel. The time discrete scheme is based on the first-order backward difference method. The memory term is approximated by the interpolating quadrature. We use Laplace transform technique to show that this interpolating quadrature scheme has a convergence rate of O ( △ t ) where △ t denotes the time step, for smooth and non-smooth initial data in homogeneous case. Special attention is that the uniform l 1 convergence property of the discretization in time is given.