Simulating Viscous Fingering with a Timespace Method and Anisotropic Mesh Adaptation

TL;DR

This paper explores a two-dimensional viscous fingering simulation using a timespace method combined with anisotropic mesh adaptation, highlighting benefits and limitations related to timestep size and numerical perturbations.

Contribution

It introduces a timespace method with anisotropic mesh adaptation for viscous fingering, demonstrating potential for reducing timesteps in practical applications.

Findings

Timespace method benefits from anisotropic elements avoiding interpolation errors.

A minimum timestep size exists due to numerical perturbations growth.

Number of timesteps can be significantly reduced for practical use.

Abstract

We report findings related to a two dimensional viscous fingering problem solved with a timespace method and anisotropic elements. Timespace methods have attracted interest for solution of time dependent partial differential equations due to the implications of parallelism in the temporal dimension, but there are also attractive features in the context of anisotropic mesh adaptation; not only are heuristics and interpolation errors avoided, but slanted elements in timespace also correspond to long and accurate timesteps, i.e. the anisotropy in timespace can be exploited. We show that our timespace method is restricted by a minimum timestep size, which is due to the growth of numerical perturbations. The lower bound on the timestep is, however, quite high, which is indicative that the number of timesteps can be reduced with several orders of magnitude for practical applications.