A Three-Level Linearized Time Integration Scheme for Tumor Simulations with Cahn-Hilliard Equations

摘要

The paper contains an analysis of a three-level linearized time integration scheme for Cahn-Hilliard equations. We start with a rigorous mixed strong/variational formulation of the appropriate initial boundary value problem taking into account the existence and uniqueness of its solution. Next we pass to the definition of two time integration schemes: the Crank-Nicolson and a three-level linearized ones. Both schemes are applied to the discrete version of Cahn-Hilliard equation obtained through the Galerkin approximation in space. We prove that the sequence of solutions of the mixed three level finite difference scheme combined with the Galerkin approximation converges when the time step length and the space approximation error decrease. We also recall the verification of the second order of this scheme and its unconditional stability with respect to the time variable. A comparative scalability analysis of parallel implementations of the schemes is also presented.