The dynamics of geometric PDEs: Surface evolution equations and a comparison with their small gradient approximations

Area of Science:

• Mathematical modeling
• Computational geometry
• Surface evolution dynamics

Background:

• Surface evolution is often modeled using 2D partial differential equations (PDEs).
• Traditional models rely on small gradient approximations, limiting their applicability to surfaces with significant curvature.
• Existing methods are often dependent on surface parametrization.

Purpose of the Study:

• To develop a method for simulating surface evolution that overcomes the limitations of the small gradient approximation.
• To introduce a parametrization-independent approach for modeling surface dynamics.
• To enable accurate simulation of surface evolution on both simple and complex geometries.

Main Methods:

• Utilizing geometric partial differential equations (PDEs) to describe surface evolution.
• Developing a novel simulation method based on local geometric properties.
• Comparing results with traditional PDEs for both small and large gradient scenarios.

Main Results:

• The proposed geometric PDE method accurately simulates surface evolution, even with large gradients.
• Results align with traditional PDEs for small gradients but diverge significantly for large gradients.
• The method demonstrates independence from surface parametrization, enhancing its versatility.

Conclusions:

• The small gradient approximation is a significant limitation in traditional surface evolution models.
• Geometric PDEs offer a more robust framework for simulating surface dynamics across diverse geometries.
• The developed method provides a powerful tool for studying phenomena like erosion and deposition, exemplified by the Kuramoto-Sivashinsky equation.