Linear Galerkin-Legendre spectral scheme for a degenerate nonlinear and nonlocal parabolic equation arising in climatology

TL;DR

This paper develops a spectral numerical scheme using Legendre polynomials for a complex degenerate nonlinear nonlocal parabolic equation in climate modeling, achieving high accuracy and linearization through extrapolation.

Contribution

The paper introduces a fully discrete spectral method with optimal accuracy for a nonlinear nonlocal climate model, utilizing Legendre basis and extrapolation for linearization.

Findings

Scheme achieves spectral accuracy in space

Method is second order in time

Numerical tests confirm theoretical accuracy

Abstract

A special place in climatology is taken by the so-called conceptual climate models. These relatively simple sets of differential equations can successfully describe single mechanisms of climate. We focus on one family of such models based on the global energy balance. This gives rise to a degenerate nonlocal parabolic nonlinear partial differential equation for the zonally averaged temperature. We construct a fully discrete numerical method that has an optimal spectral accuracy in space and second order in time. Our scheme is based on the Galerkin formulation of the Legendre basis expansion, which is particularly convenient for this setting. By using extrapolation, the numerical scheme is linear even though the original equation is nonlinear. We also test our theoretical results during various numerical simulations that support the aforementioned accuracy of the scheme.