HOLA: a High-Order Lie Advection of Discrete Differential Forms With Applications in Fluid Dynamics

McKenzie, 2007

Category: Geometric Computing

Overall Rating

2.6/5 (18/35 pts)

This paper presents HOLA, a method that marries Discrete Exterior Calculus (DEC) with high-order WENO schemes to perform Lie advection of discrete differential forms.
...the specific implementation of the interior product introduces a critical practical bottleneck through a simple Euler backtracking step, resulting in undesirable CFL-style time step limitations.
This paper does not offer a unique, actionable path for impactful modern research pursuit.
It stands more as a historical exploration of applying high-order numerical schemes to DEC operators...

This thesis presents HOLA, a method for high-order Lie advection of arbitrary discrete differential forms on discrete manifolds, combining the geometric structure of Discrete Exterior Calculus (DEC) with the high-order accuracy and shock-capturing capabilities of WENO schemes.
The potential for novel research lies particularly in the burgeoning field of Geometric Deep Learning (GDL).
HOLA provides a blueprint for such a "geometric transport layer."
Modern GPU power makes the stencil-based high-order WENO computations and the linear system solves... more feasible on larger discrete structures than in 2007.

The paper's core relies on Discrete Exterior Calculus (DEC) as the fundamental discrete framework... it has not become the ubiquitous standard for general-purpose numerical methods...
Crucially, the backtracking step for the interior product approximation uses a simple forward Euler time discretization (p. 13, 19), which... leads to an "undesirable CFL style condition" on the maximum time step.
This single point is a critical flaw for practical application.
...makes the overall method computationally inefficient for achieving numerical stability compared to alternatives available at the time.

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