Modelling of transient heat conduction with diffuse interface methods

We present a survey on different numerical interpolation schemes used for two-phase transient heat conduction problems in the context of interface capturing phase-field methods. Examples are general transport problems in the context of diffuse interface methods with a non-equal heat conductivity in normal and tangential directions to the interface. We extend the tonsorial approach recently published by Nicoli M et al (2011 Phys. Rev. E 84 1-6) to the general three-dimensional (3D) transient evolution equations. Validations for one-dimensional, two-dimensional and 3D transient test cases are provided, and the results are in good agreement with analytical and numerical reference solutions.