  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.
