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In this work, the z-transform is presented to analyze time-discrete solutionsfor Volterra integrodifferential equations (VIDEs) with nonsmooth multi-termkernels in the Hilbert space, and this class of continuous problem was firstconsidered and analyzed by Hannsgen and Wheeler (SIAM J Math Anal 15 (1984)579-594). This work discusses three cases of kernels $\beta_q(t)$ included inthe integrals for the multi-term VIDEs, from which we use correspondingnumerical techniques to approximate the solution of multi-term VIDEs indifferent cases. Firstly, for the case of $\beta_1(t), \beta_2(t) \in\mathrm{L}_1(\mathbb{R}_+)$, the Crank-Nicolson (CN) method and interpolationquadrature (IQ) rule are applied to time-discrete solutions of the multi-termVIDEs; secondly, for the case of $\beta_1(t)\in \mathrm{L}_1(\mathbb{R}_+)$ and$\beta_2(t)\in \mathrm{L}_{1,\text{loc}}(\mathbb{R}_+)$, second-order backwarddifferentiation formula (BDF2) and second-order convolution quadrature (CQ) areemployed to discretize the multi-term problem in the time direction; thirdly,for the case of $\beta_1(t), \beta_2(t)\in\mathrm{L}_{1,\text{loc}}(\mathbb{R}_+)$, we utilize the CN method andtrapezoidal CQ (TCQ) rule to approximate temporally the multi-term problem.Then for the discrete solution of three cases, the long-time global stabilityand convergence are proved based on the z-transform and certain appropriateassumptions. Furthermore, the long-time estimate of the third case is confirmedby the numerical tests.

Global behavior of temporal discretizations for Volterra integrodifferential equations with certain nonsmooth kernels