Time-Space Adaptive Method of Time Layers for the Advective Allen-Cahn Equation
This work provides a numerical method for efficiently solving a specific class of interface problems with small surface tension, but is incremental as it adapts existing techniques.
The paper develops an adaptive time-stepping method combined with a Rosenbrock integrator and interior penalty Galerkin discretization for the advective Allen-Cahn equation. Numerical tests show accuracy and efficiency for resolving sharp interfaces in convection-dominated problems.
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.