  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.
