On the Evolutionary Dynamics of the Cahn-Hilliard Equation with Cut-Off Mass Source

Abstract

We investigate the effect of cutoff logistic source on evolutionary dynamics of a generalized Cahn-Hilliard (CH) equation in this paper. It is a well-known fact that the maximum principle does not hold for the CH equation. Therefore, a generalized CH equation with logistic source may cause the negative concentration blow-up problem in finite time. To overcome this drawback, we propose the cutoff logistic source such that only the positive value greater than a given critical concentration can grow. We consider the temporal profiles of numerical results in the one-, two-, and three-dimensional spaces to examine the effect of extra mass source. Numerical solutions are obtained using a finite difference multigrid solver. Moreover, we perform numerical tests for tumor growth simulation, which is a typical application of generalized CH equations in biology. We apply the proposed cutoff logistic source term and have good results.

An Adaptive Time-Stepping Algorithm for the Allen–Cahn Equation

“…Here, the homogeneous Neumann boundary condition [35] is used. Given final time T, tolerance tol, current time t, time step Δt, and numerical solution φ n ijk at t, the RKF method is as follows.…”

On the modified of the one-dimensional Cahn-Hilliard equation with a source term

“…The existence of finite dimensional attractors was established. Numerical simulations were performed, e.g., in [1,14,18]. Our purpose in this manuscript is to obtain a similar asymptotic result for a linear time discretization of (1.1)-(1.2) with fixed time step δt > 0.…”