A new insight on positivity and contractivity of the Crank-Nicolson scheme for the heat equation

TL;DR

This paper investigates the positivity and contractivity of the Crank-Nicolson scheme for the heat equation, providing bounds and insights that enhance understanding of its qualitative properties in numerical analysis.

Contribution

It introduces a new approach using Chebyshev-like polynomials to analyze and bound positivity and contractivity, offering a comprehensive framework for these properties.

Findings

Derived necessary and sufficient bounds for positivity and contractivity.

Provided equations and intervals for bounds using bisection process.

Highlighted differences between numerical positivity and contractivity.

Abstract

In this paper we study numerical positivity and contractivity in the infinite norm of Crank-Nicolson method when it is applied to the diffusion equation with homogeneous Dirichlet boundary conditions. For this purpose, the amplification matrices are written in terms of three kinds of Chebyshev-like polynomials, and necessary and sufficient bounds to preserve the desired qualitative properties are obtained. For each spatial mesh, we provide the equations that must be solved as well as the intervals that contain these bounds; consequently, they can be easily obtained by a bisection process. Besides, differences between numerical positivity and contractivity are highlighted. This problem has also been addressed by some other authors in the literature and some known results are recovered in our study. Our approach gives a new insight on the problem that completes the panorama and that can be used to study qualitative properties for other problems.