# Kernel-based collocation methods for heat transport on evolving surfaces

## TL;DR

This paper introduces kernel-based meshless collocation algorithms for solving heat and surfactant transport PDEs on evolving surfaces, capable of handling high curvature and discontinuities.

## Contribution

It presents two novel kernel-based meshless collocation methods tailored for PDEs on evolving surfaces, including direct PDE collocation and pseudospectral surface operator approximation.

## Key findings

- Successfully solves PDEs on high-curvature surfaces
- Handles discontinuous initial conditions accurately
- Demonstrates convergence and robustness of methods

## Abstract

We propose algorithms for solving convective-diffusion partial differential equations (PDEs), which model surfactant concentration and heat transport on evolving surfaces, based on intrinsic kernel-based meshless collocation methods. The algorithms can be classified into two categories: one collocates PDEs directly and analytically, and the other approximates surface differential operators by meshless pseudospectral approaches. The former is specifically designed to handle PDEs on evolving surfaces defined by parametric equations, and the latter works on surface evolutions based on point clouds. After some convergence studies and comparisons, we demonstrate that the proposed method can solve challenging PDEs posed on surfaces with high curvatures with discontinuous initial conditions with correct physics.