Solving parabolic equations on the unit sphere via Laplace transforms and radial basis functions

TL;DR

This paper introduces a numerical method combining Laplace transforms and radial basis functions to solve parabolic equations on the sphere with high accuracy and parallel computation capabilities.

Contribution

It presents a novel approach integrating Laplace transform-based time discretization with radial basis functions for spatial approximation on the sphere.

Findings

Achieves high accuracy in numerical solutions

Provides $L_2$ error estimates for various initial data

Demonstrates effectiveness through numerical experiments

Abstract

We propose a method to construct numerical solutions of parabolic equations on the unit sphere. The time discretization uses Laplace transforms and quadrature. The spatial approximation of the solution employs radial basis functions restricted to the sphere. The method allows us to construct high accuracy numerical solutions in parallel. We establish $L_2$ error estimates for smooth and nonsmooth initial data, and describe some numerical experiments.