Adaptive time-stepping for Cahn-Hilliard-type equations with application to diffuse-interface tumor-growth models

Many tumor-growth phenomena can be considered as multiphase problems. Employing the continuum theory of mixtures, phase-field tumor-growth models can be derived with diffuse interfaces. The chosen form of the Helmholtz free-energy leads to equations of the Cahn-Hilliard type. Such nonlinear fourth-order partial-differential equations are time-dependent, and their solutions exhibit alternating fast and slow variations in time. It is therefore of prime importance to use adaptive time-stepping to efficiently simulate the entire dynamics of the system [5]. In this contribution, we consider a thermodynamically consistent four-species model of tumor growth in which the energy is non-increasing and total mass is conserved [6]. In order to inherit these two main characteristics of the system at the discrete level, we propose a gradient-stable time-stepping scheme with second-order accuracy [8]. Mixed finite elements are used for spatial discretization. For this discretization, we discuss various adaptive time-stepping strategies in time. Furthermore, we present illustrative numerical results.