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