A Semi-Lagrangian scheme on embedded manifolds using generalized local polynomial reproductions

TL;DR

This paper develops a high-order semi-Lagrangian scheme for PDEs on embedded manifolds, introducing a mesh-free remapping operator based on generalized polynomial reproduction that achieves high accuracy and stability.

Contribution

It introduces a novel mesh-free remapping operator using $ ext{l}_1$ minimization for high-order accuracy on manifolds, extending error analysis for semi-Lagrangian schemes.

Findings

The proposed scheme achieves high uniform convergence rates.

Numerical experiments validate the theoretical error estimates.

The method requires only point values, no geometric manifold information.

Abstract

We analyze rates of uniform convergence for a class of high-order semi-Lagrangian schemes for first-order, time-dependent partial differential equations on embedded submanifolds of $\mathbb{R}^d$ (including advection equations on surfaces) by extending the error analysis of Falcone and Ferretti. A central requirement in our analysis is a remapping operator that achieves both high approximation orders and strong stability, a combination that is challenging to obtain and of independent interest. For this task, we propose a novel mesh-free remapping operator based on $\ell_1$ minimizing generalized polynomial reproduction, which uses only point values and requires no additional geometric information from the manifold (such as access to tangent spaces or curvature). Our framework also rigorously addresses the numerical solution of ordinary differential equations on manifolds via projection methods. We include numerical experiments that support the theoretical results and also suggest some new directions for future research.