Efficient implementation of characteristic-based schemes on unstructured triangular grids

TL;DR

This paper improves the efficiency of characteristic-based schemes for advection in unstructured grids by enhancing characteristic location methods, enabling larger time steps and better applicability to complex geometries.

Contribution

It introduces a novel technique that leverages additional information about the advecting field to speed up characteristic foot location on unstructured grids.

Findings

Significant reduction in execution times for unstructured grid schemes.

Extended applicability of large time-step schemes to complex geometries.

Maintained stability of schemes with improved efficiency.

Abstract

Using characteristics to treat advection terms in time-dependent PDEs leads to a class of schemes, e.g., semi-Lagrangian and Lagrange-Galerkin schemes, which preserve stability under large Courant numbers, and may therefore be appealing in many practical situations. Unfortunately, the need of locating the feet of characteristics may cause a serious drop of efficiency in the case of unstructured space grids, and thus prevent the use of large time-step schemes on complex geometries. In this paper, we perform an in-depth analysis of the main recipes available for characteristic location, and propose a technique to improve the efficiency of this phase, using additional information related to the advecting vector field. This results in a clear improvement of execution times in the unstructured case, thus extending the range of applicability of large time-step schemes.