In this paper, we investigate and analyze numerical solutions for the Volterra integrodifferential equations with tempered multi-term kernels. Firstly we derive some regularity estimates of the exact solution. Then a temporal-discrete scheme is established by employing Crank-Nicolson technique and product integration (PI) rule for discretizations of the time derivative and tempered-type fractional integral terms, respectively, from which, nonuniform meshes are applied to overcome the singular behavior of the exact solution at $t=0$. Based on deduced regularity conditions, we prove that the proposed scheme is unconditionally stable and conservative, and possesses accurately temporal second-order convergence in $L_2$-norm. Numerical examples confirm the effectiveness of the proposed method.

翻译：在本文中,我们调查并分析具有温和多期内核的Volterra内分异方程式的数字解决方案。 首先,我们得出对确切解决方案的一些定期性估计。 然后,通过采用Crank-Nicolson技术和产品集成(PI)规则,分别对时间衍生物和温和型分解集成术语进行分解,从中,采用非统一的模具来克服以$t=0计算的精确解决方案的奇特行为。根据推断的常规性条件,我们证明拟议方案无条件稳定和保守,并准确掌握以$_2美元-诺尔姆计算的时序第二级趋同。 数字实例证实了拟议方法的有效性。