This paper studies a class of convolution quadratures, well-known numerical methods for calculation of convolution integrals. In contrast to the existing counterpart, which uses the linear multistep formula or Runge-Kutta method, we employ the block generalized Adams method to discretize the underlying initial value problem. Similar to the convolution quadrature method based on the linear multistep formula, the proposed method can also be implemented on an equispaced grid. In addition, the proposed approach is as stable as the convolution quadrature based on the Runge-Kutta method, which indicates that it can accurately solve a wide range of problems without becoming unstable. We provide a detailed convergence analysis for the proposed convolution quadrature method and numerically illustrate our theoretical findings for convolution integrals with smooth and weakly singular kernels.

翻译：本文研究一类卷积求积方法，这是计算卷积积分的经典数值方法。与现有基于线性多步法或龙格-库塔方法的卷积求积不同，我们采用块广义Adams方法离散底层初值问题。与基于线性多步法的卷积求积方法类似，所提方法同样可在等距网格上实现。此外，所提方法与基于龙格-库塔方法的卷积求积具有同等稳定性，表明其能在不产生不稳定的前提下精确求解广泛问题。我们对该卷积求积方法进行了详细的收敛性分析，并通过光滑核与弱奇异核的卷积积分算例，数值验证了理论结果。