The Characteristic Mapping Method for the Linear Advection of Arbitrary Sets

TL;DR

The paper introduces a novel numerical method called the Characteristic Mapping Method for efficiently transporting arbitrary sets in a velocity field by computing deformation maps and advecting sets through function composition, reducing computational costs.

Contribution

It proposes a new approach that separates short-term advection from long-term deformation storage, improving accuracy and efficiency over existing methods.

Findings

Accurate results demonstrated in 2D and 3D simulations.

Significant reduction in computational time compared to other methods.

Effective transport of multiple sets with low computational cost.

Abstract

We present a new numerical method for transporting arbitrary sets in a velocity field. The method computes a deformation mapping of the domain and advects particular sets by function composition with the map. This also allows for the transport of multiple sets at low computational cost. Our strategy is to separate the computation of short time advection from the storage and representation of long time deformation maps, employing appropriate grid resolution for each of these two parts. We show through numerical experiments that the resulting algorithm is accurate and exhibits significant reductions in computational time over other methods. Results are presented in two and three dimensions, and accuracy and efficiency are studied.