The Experts below are selected from a list of 5298 Experts worldwide ranked by ideXlab platform

Etienne Emmrich - One of the best experts on this subject based on the ideXlab platform.

Cheng Wang - One of the best experts on this subject based on the ideXlab platform.

  • a second order accurate scalar auxiliary variable sav numerical method for the square phase field crystal equation
    arXiv: Numerical Analysis, 2020
    Co-Authors: Min Wang, Qiumei Huang, Cheng Wang
    Abstract:

    In this paper we propose and analyze a second order accurate (in time) numerical scheme for the square phase field crystal (SPFC) equation, a gradient flow modeling crystal dynamics at the atomic scale in space but on diffusive scales in time. Its primary difference with the standard phase field crystal model is an introduction of the 4-Laplacian term in the free energy potential, which in turn leads to a much higher degree of nonlinearity. To make the numerical scheme linear while preserving the nonlinear energy stability, we make use of the scalar auxiliary variable (SAV) approach, in which a second order backward Differentiation Formula (BDF) is applied in the temporal stencil. Meanwhile, a direct application of the SAV method faces certain difficulties, due to the involvement of the 4-Laplacian term, combined with a derivation of the lower bound of the nonlinear energy functional. In the proposed numerical method, an appropriate decomposition for the physical energy functional is Formulated, so that the nonlinear energy part has a well-established global lower bound, and the rest terms lead to constant-coefficient diffusion terms with positive eigenvalues. In turn, the numerical scheme could be very efficiently implemented by constant-coefficient Poisson-like type solvers (via FFT), and energy stability is established by introducing an auxiliary variable, and an optimal rate convergence analysis is provided for the proposed SAV method. A few numerical experiments are also presented, which confirm the efficiency and accuracy of the proposed scheme.

  • second order semi implicit projection methods for micromagnetics simulations
    Journal of Computational Physics, 2020
    Co-Authors: Changjian Xie, Cheng Wang, Carlos J Garciacervera, Zhennan Zhou, Jingrun Chen
    Abstract:

    Abstract Micromagnetics simulations require accurate approximation of the magnetization dynamics described by the Landau-Lifshitz-Gilbert equation, which is nonlinear, nonlocal, and has a non-convex constraint, posing interesting challenges in developing numerical methods. In this paper, we propose two second-order semi-implicit projection methods based on the second-order backward Differentiation Formula and the second-order interpolation Formula using the information at previous two temporal steps. Unconditional unique solvability of both methods is proved, with their second-order accuracy verified through numerical examples in both 1D and 3D. The efficiency of both methods is compared to that of another two popular methods. In addition, we test the robustness of both methods for the first benchmark problem with a ferromagnetic thin film material from National Institute of Standards and Technology.

  • a second order energy stable backward Differentiation Formula method for the epitaxial thin film equation with slope selection
    Numerical Methods for Partial Differential Equations, 2018
    Co-Authors: Wenqiang Feng, Cheng Wang, Steven M Wise, Zhengru Zhang
    Abstract:

    In this paper, we study a novel second-order energy stable Backward Differentiation Formula (BDF) finite difference scheme for the epitaxial thin film equation with slope selection (SS). One major challenge for the higher oder in time temporal discretization is how to ensure an unconditional energy stability and an efficient numerical implementation. We propose a general framework for designing the higher order in time numerical scheme with unconditional energy stability by using the BDF method with constant coefficient stabilized terms. Based on the unconditional energy stability property, we derive an $L^\infty_h (0,T; H_{h}^2)$ stability for the numerical solution and provide an optimal the convergence analysis. To deal with the 4-Laplacian solver in an $L^{2}$ gradient flow at each time step, we apply an efficient preconditioned steepest descent algorithm and preconditioned nonlinear conjugate gradient algorithm to solve the corresponding nonlinear system. Various numerical simulations are present to demonstrate the stability and efficiency of the proposed schemes and slovers.

Jingrun Chen - One of the best experts on this subject based on the ideXlab platform.

  • second order semi implicit projection methods for micromagnetics simulations
    Journal of Computational Physics, 2020
    Co-Authors: Changjian Xie, Cheng Wang, Carlos J Garciacervera, Zhennan Zhou, Jingrun Chen
    Abstract:

    Abstract Micromagnetics simulations require accurate approximation of the magnetization dynamics described by the Landau-Lifshitz-Gilbert equation, which is nonlinear, nonlocal, and has a non-convex constraint, posing interesting challenges in developing numerical methods. In this paper, we propose two second-order semi-implicit projection methods based on the second-order backward Differentiation Formula and the second-order interpolation Formula using the information at previous two temporal steps. Unconditional unique solvability of both methods is proved, with their second-order accuracy verified through numerical examples in both 1D and 3D. The efficiency of both methods is compared to that of another two popular methods. In addition, we test the robustness of both methods for the first benchmark problem with a ferromagnetic thin film material from National Institute of Standards and Technology.

Carlos J Garciacervera - One of the best experts on this subject based on the ideXlab platform.

  • second order semi implicit projection methods for micromagnetics simulations
    Journal of Computational Physics, 2020
    Co-Authors: Changjian Xie, Cheng Wang, Carlos J Garciacervera, Zhennan Zhou, Jingrun Chen
    Abstract:

    Abstract Micromagnetics simulations require accurate approximation of the magnetization dynamics described by the Landau-Lifshitz-Gilbert equation, which is nonlinear, nonlocal, and has a non-convex constraint, posing interesting challenges in developing numerical methods. In this paper, we propose two second-order semi-implicit projection methods based on the second-order backward Differentiation Formula and the second-order interpolation Formula using the information at previous two temporal steps. Unconditional unique solvability of both methods is proved, with their second-order accuracy verified through numerical examples in both 1D and 3D. The efficiency of both methods is compared to that of another two popular methods. In addition, we test the robustness of both methods for the first benchmark problem with a ferromagnetic thin film material from National Institute of Standards and Technology.

Yves Bourgault - One of the best experts on this subject based on the ideXlab platform.

  • efficient second order semi implicit finite element method for fourth order nonlinear diffusion equations
    Computer Physics Communications, 2021
    Co-Authors: Sana Keita, Abdelaziz Beljadid, Yves Bourgault
    Abstract:

    Abstract We focus here on a class of fourth-order parabolic equations that can be written as a system of second-order equations by introducing an auxiliary variable. We design a novel second-order fully discrete mixed finite element method to approximate these equations. In our approach, we propose new techniques using the second-order backward Differentiation Formula for the time derivative and a special technique for the approximation of nonlinear terms. The use of the proposed technique for nonlinear terms makes the developed numerical scheme efficient in terms of computational cost since the proposed method only deals with a linear system at each time step and no iterative resolution is needed. A numerical convergence study is performed using the method of manufactured and analytical solutions of the system where we investigate different boundary conditions. With respect to the spatial discretization, convergence rates are found to at least match a priori error estimates available for linear problems. The convergence analysis is completed with an investigation of the temporal discretization where we numerically demonstrate the second-order time-accuracy of the proposed scheme using the method of reference solution. We present a series of numerical tests to demonstrate the efficiency and robustness of the proposed scheme.