Approximate solutions to second-order parabolic equations: Evolution systems and discretization

Wen Cheng; Anna L. Mazzucato; Victor Nistor

doi:10.3934/dcdss.2022158

Article Contents

2022, Volume 15, Issue 12: 3571-3602. Doi: 10.3934/dcdss.2022158

This issue Previous Article Soliton solutions for quasilinear modified Schrödinger equations in applied sciences Next Article On some qualitative aspects for doubly nonlocal equations

Approximate solutions to second-order parabolic equations: Evolution systems and discretization

1.
Equity Derivative Quantitative Strategies, Credit Suisse, 11 Madison Avenue, New York, NY 10010, USA
2.
Penn State University, Mathematics Department, University Park, PA 16802, USA
3.
Université' de Lorraine, 3, rue Augustin Fresnel, 57000 Metz, France

^*Corresponding author: Anna L. Mazzucato
^*Corresponding author: Anna L. Mazzucato

In loving memory of Rosa Maria (Rosella) Mininni

Received: October 2021

Revised: May 2022

Early access: September 2022

Published: December 2022

A.M. was partially supported by the US National Science Foundation grant DMS-1909103

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Abstract

We study the discretization of a linear evolution partial differential equation when its Green's function is known or well approximated. We provide error estimates both for the spatial approximation and for the time stepping approximation. We show that, in fact, an approximation of the Green function is almost as good as the Green function itself. For suitable time-dependent parabolic equations, we explain how to obtain good, explicit approximations of the Green function using the Dyson-Taylor commutator method that we developed in J. Math. Phys. 51 (2010), n. 10, 103502 (reference [15]). This approximation for short time, when combined with a bootstrap argument, gives an approximate solution on any fixed time interval within any prescribed tolerance.

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Mathematics Subject Classification: Primary: 35K45, 35K08; Secondary: 65M80.

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