Stabilized second‐order convex splitting schemes for Cahn–Hilliard models with application to diffuse‐interface tumor‐growth models

Abstract

We present unconditionally energy-stable second-order time-accurate schemes for diffuse-interface (phase-field) models; in particular, we consider the Cahn-Hilliard equation and a diffuse-interface tumor-growth system consisting of a reactive Cahn-Hilliard equation and a reaction-diffusion equation. The schemes are of the Crank-Nicolson type with a new convex-concave splitting of the free energy and an artificial-diffusivity stabilization. The case of nonconstant mobility is treated using extrapolation. For the tumor-growth system, a semi-implicit treatment of the reactive terms and additional stabilization are discussed. For suitable free energies, all schemes are linear. We present numerical examples that verify the second-order accuracy, unconditional energy-stability, and superiority compared with their first-order accurate variants.

An energy-stable time-integrator for phase-field models

“…The diffuse interface is introduced through an energetic variational procedure that results in a thermodynamically consistent coupling. This consistency gives the method solid mathematical and physical footings, and explains why applications range from spinodal decomposition of immiscible binary mixtures [7,21,61], tumor angiogenesis [58,59], wetting [14] and elasto-capillarity [5,53], image processing [40] to water infiltration in porous media [22]. Another advantage of the phase-field method over sharp interface descriptions comes from the fact that under appropriate assumptions [6], a diffuse-interface description can asymptotically converge to its sharp-interface counterpart by decreasing the interfacial thickness.…”

“…To prove second order accuracy in time for our scheme, we compute the next time-step approximation via the scheme applied to the exact solution and compare the result to Taylor expansions. A similar procedure was performed in [61] in the context of the Cahn-Hilliard equation and in [54] in the context of the phase-field crystal equation. Using equations (9)- (10), and reorganizing the splitting into one equation, we have that…”

Computational Phase‐Field Modeling

“…Stabilization Most second-order linear schemes suffer from conditional stability and/or conditional solvability. As proposed by (Wu et al, 2014) these issues can be stabilized if W satisfies (A2). For example, the stabilized semi-implicit scheme is…”

“…4.1.3: Note that we used first-order extrapolation on the nonlinearity caused by g(φ h ). This scheme can be shown to be unconditionally energy stable so that large time-steps can be taken (Wu et al, 2014).…”

Numerical Methods for Solving the Cahn–Hilliard Equation and Its Applicability to Related Energy-Based Models

“…Furthermore, we introduce a new adaptive time step algorithm based on the numerical dissipation introduced in the discrete energy law in each time step. Finally, in the context of tumor growth models, in [67] a linear scheme is presented which is unconditionally energy-stable using a modified formulation of the potential (1.2) and splitting the potential to relax the growth of the potential F (φ) when |φ| > 1.…”