Isogeometric analysis of the Cahn–Hilliard phase-field model

Linear, first and second-order, unconditionally energy stable numerical schemes for the phase field model of homopolymer blends

Stable and efficient finite-difference nonlinear-multigrid schemes for the phase field crystal equation

Provably unconditionally stable, second-order time-accurate, mixed variational methods for phase-field models

In this work, we will analyze a class of large time-stepping methods for the Cahn-Hilliard equation. The equation is discretized by Fourier spectral method in space and semi-implicit schemes in time. For first-order semi-implicit scheme, the stability and convergence properties are investigated based on an energy approach. Here stability means that the decay of energy is preserved. The numerical experiments are used to demonstrate the effectiveness of the large time-stepping approaches.

https://www.math.hkbu.edu.hk/~ttang/Papers/tang_ANM.pdf

On large time-stepping methods for the Cahn--Hilliard equation

A class of stable spectral methods for the Cahn-Hilliard equation

A spectral Galerkin method in the spatial discretization is analyzed to solve the Cahn-Hilliard equation. Existence, uniqueness, and stabilities for both the exact solution and the approximate solution are given. Using the theory and technique of a priori estimate for the partial differential equation, we obtained the convergence of the spectral Galerkin method and the error estimate between the approximate solution uN(t) and the exact solution u(t). © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008

Stability and convergence of the spectral Galerkin method for the Cahn‐Hilliard equation

We consider the drift-diffusion (DD) model of one dimensional semiconductor devices, which is a system involving not only first derivative convection terms but also second derivative diffusion terms and a coupled Poisson potential equation. Optimal error estimates are obtained for both the semi-discrete and fully discrete local discontinuous Galerkin (LDG) schemes with smooth solutions. In the fully discrete scheme, we couple the implicit-explicit (IMEX) time discretization with the LDG spatial discretization, in order to allow larger time steps and to save computational cost. The main technical difficulty in the analysis is to treat the inter-element jump terms which arise from the discontinuous nature of the numerical method and the nonlinearity and coupling of the models. A simulation is also performed to validate the analysis.

Analysis of the local discontinuous Galerkin method for the drift-diffusion model of semiconductor devices

In this paper we continue our effort in [Y. Liu, C.-W. Shu, J. Comput. Electron. 3 (2004) 263] for developing local discontinuous Galerkin (LDG) finite element methods to discretize moment models in device simulations. We consider various hydrodynamic (HD) and energy transport (ET) models, which involve not only first derivative convection terms but also second derivative diffusion (heat conduction) terms and a Poisson potential equation. The convection-diffusion system is discretized by the local discontinuous Galerkin (LDG) method. The potential equation for the electric field is also discretized by the LDG method, thus the numerical tool is based on a unified discontinuous Galerkin methodology for different components. We simulate different moment models and different devices to demonstrate the robustness of the algorithm, and also assess the performance of the algorithm with different orders of accuracy. A two-dimensional simulation is also performed for a MESFET device, producing results in agreement with that obtained by the essentially non-oscillatory (ENO) finite difference method.

Yunxian Liu

Papers

On large time-stepping methods for the Cahn--Hilliard equation

A class of stable spectral methods for the Cahn-Hilliard equation

Stability and convergence of the spectral Galerkin method for the Cahn‐Hilliard equation

Analysis of the local discontinuous Galerkin method for the drift-diffusion model of semiconductor devices

Local discontinuous Galerkin methods for moment models in device simulations: Performance assessment and two-dimensional results