This paper studies an extension of the classical convolution quadrature, a well-known numerical method 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 high-order method 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：本文研究经典卷积求积方法（一种用于计算卷积积分的著名数值方法）的扩展形式。与现有方法采用线性多步公式或龙格-库塔方法不同，我们使用块广义Adams方法对底层初值问题进行离散化。与基于线性多步公式的卷积求积方法类似，所提方法也可在等距网格上实现。此外，该高阶方法具有与基于龙格-库塔方法的卷积求积相当的稳定性，表明其能在保持稳定性的前提下精确求解广泛问题。我们为所提卷积求积方法提供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数值算例验证了理论发现。