In this work, the z-transform is presented to analyze time-discrete solutions for Volterra integrodifferential equations (VIDEs) with nonsmooth multi-term kernels in the Hilbert space, and this class of continuous problem was first considered 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 in the integrals for the multi-term VIDEs, from which we use corresponding numerical techniques to approximate the solution of multi-term VIDEs in different cases. Firstly, for the case of $\beta_1(t), \beta_2(t) \in \mathrm{L}_1(\mathbb{R}_+)$, the Crank-Nicolson (CN) method and interpolation quadrature (IQ) rule are applied to time-discrete solutions of the multi-term VIDEs; 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 backward differentiation formula (BDF2) and second-order convolution quadrature (CQ) are employed 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 and trapezoidal CQ (TCQ) rule to approximate temporally the multi-term problem. Then for the discrete solution of three cases, the long-time global stability and convergence are proved based on the z-transform and certain appropriate assumptions. Furthermore, the long-time estimate of the third case is confirmed by the numerical tests.

翻译：本文利用z变换分析Hilbert空间中具有非光滑多核函数的Volterra积分微分方程（VIDEs）的时间离散解，此类连续问题最早由Hannsgen和Wheeler（SIAM J Math Anal 15 (1984) 579-594）进行研究和分析。本文讨论了多核VIDEs积分中包含的核函数$\beta_q(t)$的三种情形，并针对不同情形采用相应数值技术逼近多核VIDEs的解。首先，针对$\beta_1(t), \beta_2(t) \in \mathrm{L}_1(\mathbb{R}_+)$的情形，采用Crank-Nicolson（CN）方法和插值求积（IQ）规则对多核VIDEs进行时间离散求解；其次，针对$\beta_1(t)\in \mathrm{L}_1(\mathbb{R}_+)$且$\beta_2(t)\in \mathrm{L}_{1,\text{loc}}(\mathbb{R}_+)$的情形，采用二阶向后微分公式（BDF2）和二阶卷积求积（CQ）在时间方向离散多核问题；最后，针对$\beta_1(t), \beta_2(t)\in \mathrm{L}_{1,\text{loc}}(\mathbb{R}_+)$的情形，采用CN方法和梯形卷积求积（TCQ）规则对多核问题进行时间近似。随后，基于z变换和若干适当假设，证明了三种情形下离散解的长时间全局稳定性和收敛性。此外，通过数值实验验证了第三种情形的长时间估计结果。