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Related Experiment Video

Updated: Aug 30, 2025

Investigating the Three-dimensional Flow Separation Induced by a Model Vocal Fold Polyp

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Published on: February 3, 2014

Meshfree Semi-Lagrangian Methods for Solving Surface Advection PDEs.

Argyrios Petras¹, Leevan Ling², Steven J Ruuth³

¹Johann Radon Institute for Computational and Applied Mathematics (RICAM), Altenbergerstrasse 69, 4040 Linz, Austria.

Journal of Scientific Computing

|August 29, 2022

Summary

This summary is machine-generated.

This study introduces meshfree semi-Lagrangian methods for advection problems on surfaces. These methods are proven robust for complex simulations, including pattern formation.

Keywords:

Closest point method Radial basis functions Semi-Lagrangian method Surface conservation laws

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Area of Science:

Computational mathematics
Numerical analysis
Surface PDEs

Background:

Solving advection problems on smooth, closed surfaces with solenoidal velocity fields presents unique challenges.
Existing meshfree semi-Lagrangian methods require adaptation for surface-based computations.

Purpose of the Study:

To develop and analyze meshfree semi-Lagrangian methods for advection problems on surfaces.
To establish convergence theories for surface-based numerical methods.
To demonstrate the adaptability of these methods for complex scientific computing tasks.

Main Methods:

Analysis of an embedding equation for surface-based semi-Lagrangian methods.
Application of standard bulk domain convergence theories to surface counterparts.
Implementation of methods on point clouds for practical application.

Main Results:

Existence of an embedding equation enabling existing semi-Lagrangian methods for surface problems.
Adaptation of bulk domain convergence theories for surface computations.
Successful implementation and verification of convergence rates on point clouds.

Conclusions:

The proposed meshfree semi-Lagrangian methods are effective for advection on surfaces.
These methods serve as a robust foundation for more complex problems like pattern formation.
The work bridges the gap between theoretical analysis and practical implementation in scientific computing.