A second order time-stepping scheme for parabolic interface problems with moving interfaces

¹ Institute of Applied Mathematics, Heidelberg University, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany.
stefan.frei@iwr.uni-heidelberg.de
² Universität Magdeburg, Universitätsplatz 2, 39106 Magdeburg, Germany.
thomas.richter@ovgu.de

Received: 23 March 2016
Revised: 29 June 2016
Accepted: 12 November 2016

Abstract

We present a second order time-stepping scheme for parabolic problems on moving domains and interfaces. The diffusion coefficient is discontinuous and jumps across an interior interface. This causes the solution to have discontinuous derivatives in space and time. Without special treatment of the interface, both spatial and temporal discretization will be sub-optimal. For such problems, we develop a time-stepping method, based on a cG(1) Eulerian space-time Galerkin approach. We show −both analytically and numerically− second order convergence in time. Key to gaining the optimal order of convergence is the use of space-time test- and trial-functions, that are aligned with the moving interface. Possible applications are multiphase flow or fluid-structure interaction problems.