Maximal regularity for time-stepping schemes arising from convolution quadrature of non-local in time equations

Carlos Lizama; Marina Murillo-Arcila

doi:10.3934/dcds.2022032

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2022, Volume 42, Issue 8: 3787-3807. Doi: 10.3934/dcds.2022032

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Maximal regularity for time-stepping schemes arising from convolution quadrature of non-local in time equations

Carlos Lizama^1, , and
Marina Murillo-Arcila^2,

1.
Universidad de Santiago de Chile, Departamento de Matemática y Ciencia de la Computación, Las Sophoras 173, Estación Central, Santiago, Chile
2.
Universitat Politècnica de València, Instituto Universitario de Matemática Pura y Aplicada (IUMPA), Camino de Vera s/n, 46022 València, Spain

^* Corresponding author: Carlos Lizama
^* Corresponding author: Carlos Lizama

Received: September 2021

Revised: February 2022

Early access: March 2022

Published: August 2022

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Abstract

We study discrete time maximal regularity in Lebesgue spaces of sequences for time-stepping schemes arising from Lubich's convolution quadrature method. We show minimal properties on the quadrature weights that determines a wide class of implicit schemes. For an appropriate choice of the weights, we are able to identify the $ \theta $-method as well as the backward differentiation formulas and the $ L1 $-scheme. Fractional versions of these schemes, some of them completely new, are also shown, as well as their representation by means of the Grünwald–Letnikov fractional order derivative. Our results extend and improve some recent results on the subject and provide new insights on the basic nature of the weights that ensure maximal regularity.

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Mathematics Subject Classification: Primary: 65R20; Secondary: 65M15, 26A33, 45K05.

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