Time-Space Adaptive Method of Time Layers for the Advective Allen-Cahn Equation

TL;DR

This paper introduces an adaptive time-layer method combining Rosenbrock and symmetric interior penalty Galerkin techniques to efficiently solve the advective Allen-Cahn equation with sharp interface layers.

Contribution

It presents a novel adaptive time-layer approach with specific numerical schemes for improved accuracy and efficiency in convection-dominated interface problems.

Findings

Demonstrates high accuracy in resolving sharp interface layers.

Shows efficiency in handling convection-dominated problems.

Validates the method through numerical simulations.

Abstract

We develop an adaptive method of time layers with a linearly implicit Rosenbrock method as time integrator and symmetric interior penalty Galerkin method for space discretization for the advective Allen-Cahn equation with non-divergence-free velocity fields. Numerical simulations for convection dominated problems demonstrate the accuracy and efficiency of the adaptive algorithm for resolving the sharp layers occurring in interface problems with small surface tension.