# A non-uniform discretization of stochastic heat equations with multiplicative noise on the unit sphere

## TL;DR

This paper develops a spectral and implicit Euler discretization method for stochastic heat equations with multiplicative noise on the sphere, validated through numerical experiments related to Earth's temperature data.

## Contribution

It introduces a non-uniform temporal discretization scheme combined with spectral spatial discretization for stochastic heat equations on the sphere.

## Key findings

- Effective discretization method demonstrated through numerical experiments.
- Applicable to Earth's temperature data analysis.
- Improved handling of multiplicative noise in spherical stochastic PDEs.

## Abstract

We investigate a discretization of a class of stochastic heat equations on the unit sphere with multiplicative noises. A spectral method is used for the spatial discretization and the truncation of the Wiener process, while an implicit Euler scheme with non-uniform steps is used for the temporal discretization. Some numerical experiments inspired by Earth's surface temperature data analysis GISTEMP provided by NASA are given.