Abstract : The phase field method is a powerful numerical method to solve moving boundary problems appearing in Materials Science and Engineering. Phase field theories are parameterized by a set of physically motivated variables and their governing equations. This mini-symposium will bring together numerical analysts and computational scientists working on phase field methods to present their recent advances in algorithm designs and applications of phase field methods. The main purposes of this mini-symposium are to review the current status, identify problems and future directions, and to promote phase field methods to a wider scientific and engineering community.
[04343] Energy stability of variable step higher order ETD-MS scheme for gradient flows
Author(s) :
Xiaoming Wang (Missouri University of Science and Technology)
Abstract : We present a family of ETD-MS based variable-step higher order numerical schemes for a family of gradient flows. We demonstrate the energy stability of this family of numerical algorithms. Numerical examples will be provided to show the effectiveness of the schemes.
[04034] Energy Dissipation of Time-Fractional Phase-Field Equations: Analysis and Numerical methods
Author(s) :
Jiang Yang (Southern University of Science and Technology)
Abstract : There exists a well defined energy dissipation law for classical phase-field equations, i.e., the energy is non-increasing with respect to time. However, it is not clear how to extend the energy definition to time-fractional phase-field equations so that the corresponding dissipation law is still satisfied. In this work, we will try to settle this problem for phase-field equations with Caputo time-fractional derivative, by defining a nonlocal energy as an averaging of the classical energy with a time-dependent weight function. To deal with this, we propose a new technique on judging the positive definiteness of a symmetric function, that is derived from a special Cholesky decomposition. Then, the nonlocal energy is proved to be dissipative under a simple restriction of the weight function. Within the same framework, the time fractional derivative of classical energy for time-fractional phase-field models can be proved to be always nonpositive. At the discrete level, a fast L2-1$_\sigma$ method on general nonuniform meshes is employed. The global-in-time $H^1$-stability is established via the same framework.
[02973] A Spectral Element in Time Method for Nonlinear Gradient Systems
Author(s) :
Shiqin Liu (University of Chinese Academy of Sciences)
Haijun Yu (Academy of Mathematics and Systems Science, Chinese Academy of Sciences)
Abstract : We present a spectral element in time method for large scale nonlinear gradient systems, with the phase-field Allen-Cahn equation as an example. Different to commonly-used spectral in time methods that employ Petrov-Galerkin or weighted Galerkin approximations, the present method employs a natural variation Galerkin form that maintains volume conservation and energy dissipation property of the continuous dynamical systems. Explicit extrapolation is applied to handle the nonlinear term, which makes the method efficient. The explicit method can be improved by a few Picard iterations to obtain superconvergence. Numerical experiments confirm that the method outperforms the popular BDF4 scheme and the ETD-RK4 method.
[02987] Energy stability and error analysis of high-precision algorithms for two-phase incompressible flows
Author(s) :
Xiaoli Li (Shandong University)
Abstract : In this talk, we will first present several efficient and high-precision schemes for the two-phase incompressible flows. These schemes are linear, decoupled and only require solving a sequence of Poisson type equations at each time step. We carry out a rigorous error analysis for the first-order scheme, establishing optimal convergence rate for all relevant functions in different norms. Next we shall discuss the consistent splitting GSAV approach for the Navier-Stokes equations and carry out theoretical analysis.
[03070] NONLOCAL CAHN-HILLIARD TYPE MODEL FOR IMAGE INPAINTING
Author(s) :
Majdi AZAIEZ (Bordeaux INP & I2M UMR 5295)
Dandan JIANG (Xiamen University)
Chuanju Xu (Xiamen University )
Alain Miranville (Poitier University)
Abstract : In this talk, we propose a Cahn-Hilliard type nonlocal model for image inpainting
which is equipped with a nonlocal diffusion operator for image inpainting. For its approximation we use the modified convex splitting method for the temporal discretization with the non-local diffusion term treated implicitly, and the fidelity term treated explicitly. Spatial discretization is performed by the Fourier collocation method. We will provide several numerical experiments to assess the efficiency of our method.
[03100] A Variety of Gradient Flows: Modeling and Numerical Methods
Author(s) :
Chuanju Xu (Xiamen University)
Abstract : In this talk I will discuss a variety of gradient flow models for multi-phase problems,
derived from an energy variational formulation. The models includes fractional differential equations, equations describing the interfacial dynamics of immiscible and incompressible two-phase fluids,
dendritic crystal growth model, thermal phase change problems etc.
The talk starts with a review of the models and numerical methods for these models.
Then a new class of time-stepping schemes will be discussed.
