Phase-field model and its splitting numerical scheme for tissue growth

We consider phase-field models and associated numerical methods for tissue growth. The model consists of the Cahn-Hilliard equation with a source term. In order to solve the equations accurately and efficiently, we propose a hybrid method based on an operator splitting method. First, we solve the contribution from the source term analytically and redistribute the increased mass around the tissue boundary position. Subsequently, we solve the Cahn-Hilliard equation using the nonlinearly gradient stable numerical scheme to make the interface transition profile smooth. We then perform various numerical experiments and find that there is a good agreement when these computational results are compared with analytic solutions.