We present a novel generalized convolution quadrature method that accurately approximates convolution integrals. During the late 1980s, Lubich introduced convolution quadrature techniques, which have now emerged as a prevalent methodology in this field. However, these techniques were limited to constant time stepping, and only in the last decade generalized convolution quadrature based on the implicit Euler and Runge-Kutta methods have been developed, allowing for variable time stepping. In this paper, we introduce and analyze a new generalized convolution quadrature method based on the trapezoidal rule. Crucial for the analysis is the connection to a new modified divided difference formula that we establish. Numerical experiments demonstrate the effectiveness of our method in achieving highly accurate and reliable results.

翻译：我们提出了一种新的广义卷积求积方法，能够精确逼近卷积积分。20世纪80年代末，Lubich引入了卷积求积技术，该技术现已成为该领域的主流方法。然而，这些技术仅限于恒定时间步长，直到近十年才基于隐式欧拉法和龙格-库塔法发展出允许变时间步长的广义卷积求积方法。本文提出并分析了一种基于梯形法则的新广义卷积求积方法。我们建立的修正差商公式是其分析的关键。数值实验表明，该方法能够获得高精度且可靠的结果。