This chapter surveys recent numerical advances in the phase field method for geometric surface evolution and related geometric nonlinear partial differential equations (PDEs). Instead of describing technical details of various numerical methods and their analyses, the chapter presents a holistic overview about the main ideas of phase field modelling, its mathematical foundation, and relationships between the phase field formalism and other mathematical formalisms for geometric moving interface problems, as well as the current state of the art of numerical approximations of various phase field models with an emphasis on discussing the main ideas of numerical analysis techniques. The chapter also reviews recent development on adaptive grid methods and various applications of the phase field modelling and their numerical methods in materials science, fluid mechanics, biology and image science.

The phase field method for geometric moving interfaces and their numerical approximations

The ubiquity of semilinear parabolic equations is clear from their numerous applications ranging from physics and biology to materials and social sciences. In this paper, we consider a practically ...

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Maximum bound principles for a class of semilinear parabolic equations and exponential time-differencing schemes

We study (time) fractional Allen–Cahn and Cahn–Hilliard phase-field models to account for the anomalously subdiffusive transport behavior in heterogeneous porous materials or memory effect of certain materials. We develop an efficient finite difference scheme and a Fourier spectral scheme to effectively treat the significantly increased memory requirement and computational complexity, which arise due to the nonlocal behavior of the time-fractional models. For time fractional Cahn–Hilliard model, we observe from the numerical results that the bigger the fractional order α is, the faster the energy decays. However, for time fractional Allen–Cahn model, we derived an opposite conclusion. Moreover, we also study the coarsening dynamics for time fractional Cahn–Hilliard model, numerical results reveal that the scaling law for the energy decays as O ( t − α 3 ) , which is consistent with the well-known result O ( t − 1 3 ) for integer order Cahn–Hilliard model.

Time-fractional Allen–Cahn and Cahn–Hilliard phase-field models and their numerical investigation

A second-order and nonuniform time-stepping maximum-principle preserving scheme for time-fractional Allen-Cahn equations

The ubiquity of semilinear parabolic equations has been illustrated in their numerous applications ranging from physics, biology, to materials and social sciences. In this paper, we consider a practically desirable property for a class of semilinear parabolic equations of the abstract form $u_t=\mathcal{L}u+f[u]$ with $\mathcal{L}$ being a linear dissipative operator and $f$ being a nonlinear operator in space, namely a time-invariant maximum bound principle, in the sense that the time-dependent solution $u$ preserves for all time a uniform pointwise bound in absolute value imposed by its initial and boundary conditions. We first study an analytical framework for some sufficient conditions on $\mathcal{L}$ and $f$ that lead to such a maximum bound principle for the time-continuous dynamic system of infinite or finite dimensions. Then, we utilize a suitable exponential time differencing approach with a properly chosen generator of contraction semigroup to develop first- and second-order accurate temporal discretization schemes, that satisfy the maximum bound principle unconditionally in the time-discrete setting. Error estimates of the proposed schemes are derived along with their energy stability. Extensions to vector- and matrix-valued systems are also discussed. We demonstrate that the abstract framework and analysis techniques developed here offer an effective and unified approach to study the maximum bound principle of the abstract evolution equation that cover a wide variety of well-known models and their numerical discretization schemes. Some numerical experiments are also carried out to verify the theoretical results.

Maximum bound principles for a class of semilinear parabolic equations and exponential time differencing schemes

We consider numerical methods for solving the fractional-in-space Allen–Cahn equation which contains small perturbation parameters and strong nonlinearity. A standard fully discretized scheme for this equation is considered, namely, using the conventional second-order Crank–Nicolson scheme in time and the second-order central difference approach in space. For the resulting nonlinear scheme, we propose a nonlinear iteration algorithm, whose unique solvability and convergence can be proved. The nonlinear iteration can avoid inverting a dense matrix with only $$\mathcal {O}(N\log N)$$
 computation complexity. One major contribution of this work is to show that the numerical solutions satisfy discrete maximum principle under reasonable time step constraint. Based on the maximum stability, the nonlinear energy stability for the fully discrete scheme is established, and the corresponding error estimates are investigated. Numerical experiments are performed to verify the theoretical results.

Tianliang Hou

Papers

Numerical Analysis of Fully Discretized Crank–Nicolson Scheme for Fractional-in-Space Allen–Cahn Equations