Generalized convolution quadrature based on the trapezoidal rule

Abstract We present a novel generalized convolution quadrature (gCQ) 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 gCQ 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 gCQ 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.