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 possesses accurately temporal second-order convergence in $L_2$-norm. Numerical examples confirm the effectiveness of the proposed method.

翻译：本文研究并分析了带多核缓变系数Volterra积分微分方程的数值解。首先推导了精确解的正则性估计。随后，通过分别采用Crank-Nicolson技术对时间导数和乘积积分（PI）规则对缓变型分数阶积分项进行离散化，建立了时间离散格式，并应用非均匀网格克服精确解在$t=0$处的奇异行为。基于推导的正则性条件，证明了所提格式无条件稳定，且在$L_2$范数下具有精确的时间二阶收敛性。数值算例验证了所提方法的有效性。