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

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.