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：本文提出利用z变换分析Hilbert空间中具有非光滑多核的Volterra积分微分方程(VIDEs)的时间离散解，该类连续问题最早由Hannsgen和Wheeler (SIAM J Math Anal 15 (1984) 579-594) 提出并分析。本文针对多核VIDEs积分中出现的三种核函数$\beta_q(t)$情形，采用相应数值技术近似不同情形下的多核VIDEs解。首先，对于$\beta_1(t), \beta_2(t) \in \mathrm{L}_1(\mathbb{R}_+)$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，对于$\beta_1(t)\in \mathrm{L}_1(\mathbb{R}_+)$和$\beta_2(t)\in \mathrm{L}_{1,\text{loc}}(\mathbb{R}_+)$情形，采用二阶向后差分公式(BDF2)和二阶卷积求积(CQ)对多核问题在时间方向进行离散；再次，对于$\beta_1(t), \beta_2(t)\in \mathrm{L}_{1,\text{loc}}(\mathbb{R}_+)$情形，利用CN方法和梯形CQ (TCQ)规则对多核问题进行时间近似。随后，基于z变换和适当假设，证明三种情形下离散解的长期全局稳定性与收敛性。最后，通过数值实验验证第三种情形的长期估计。