Generalized convolution quadrature based on the trapezoidal rule

Lehel Banjai; Matteo Ferrari

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Generalized convolution quadrature based on the trapezoidal rule

Lehel Banjai, Matteo Ferrari

Numerical Analysis · Submitted 18 May 2023

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Numerical experiments demonstrate the effectiveness of our method in achieving highly accurate and reliable results.

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arXiv listing abstract v1. Results and the main claim are underlined; show every role.

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.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We present a novel generalized convolution quadrature method that accurately approximates convolution integrals.	contribution 1.00	0.94	0.57	asserted 1.00
s1	During the late 1980s, Lubich introduced convolution quadrature techniques, which have now emerged as a prevalent methodology in this field.	background 1.00	0.03	0.02	asserted 1.00
s2	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.	background 0.56	0.05	0.04	asserted 1.00
s3	In this paper, we introduce and analyze a new generalized convolution quadrature method based on the trapezoidal rule.	contribution 1.00	0.94	0.22	asserted 0.99
s4	Crucial for the analysis is the connection to a new modified divided difference formula that we establish.	method 0.56	0.93	0.07	asserted 1.00
s5	Numerical experiments demonstrate the effectiveness of our method in achieving highly accurate and reliable results.	result 0.98	0.94	0.53	asserted 1.00

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Original abstract

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.

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2 arXiv papers cite this in 3 sentences

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Cites as background 1 paper

Even if in literature there are various choices of high-order CQ based on Runge-Kutta methods (see e.g [ref] ), we focus here on CQ based on A-stable linear multistep methods (see [ref] ), and thus restricted by the Dahlquist's barrier to second order methods.

Runge-Kutta convolution quadrature based on Gauss methods · Convolution quadrature for hyperbolic symbols · 0.81

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