Approximate Solutions to Second-Order Parabolic Equations: evolution systems and discretization

TL;DR

This paper presents a method for discretizing second-order parabolic equations using Green function approximations, providing error estimates and a new Dyson-Taylor commutator technique for explicit Green function approximation.

Contribution

It introduces a novel approach to approximate Green functions for parabolic equations, enabling accurate discretization and solution within prescribed tolerances.

Findings

Green function approximation is nearly as effective as the exact Green function.

Error estimates are established for spatial and temporal discretizations.

The Dyson-Taylor commutator method provides explicit Green function approximations for short times.

Abstract

We study the discretization of a linear evolution partial differential equation when its Green function is known. 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 (DTCM) that we developed in J. Math. Phys. (2010). This approximation for short time, when combined with a bootstrap argument, gives an approximate solution on any fixed time interval within any prescribed tolerance.