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1. SECOND ORDER, LINEAR, AND UNCONDITIONALLY ENERGY STABLE SCHEMES FOR A HYDRODYNAMIC MODEL OF SMECTIC-A LIQUID CRYSTALS

   Chen, R ; Yang, X ; Zhang, H. ( November 2017 , SIAM journal on scientific computing)

   In this paper, we consider the numerical approximations for a hydrodynamical model of smectic-A liquid crystals. The model, derived from the variational approach of the modified Oseen– Frank energy, is a highly nonlinear system that couples the incompressible Navier–Stokes equations and a constitutive equation for the layer variable. We develop two linear, second order time marching schemes based on the Invariant Energy Quadratization method for nonlinear terms in the constitutive equation, the projection method for the Navier–Stokes equations, and some subtle implicit-explicit treatments for the convective and stress terms. Moreover, we prove the well-posedness of the linear system and their unconditionally energy stabilities rigorously. Various numerical experiments are presented to demonstrate the stability and the accuracy of the numerical schemes in simulating the dynamics under shear flow and the magnetic field. 

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2. On a novel full decoupling, linear, second‐order accurate, and unconditionally energy stable numerical scheme for the anisotropic phase‐field dendritic crystal growth model

   https://doi.org/10.1002/nme.6697  

   Yang, Xiaofeng ( April 2021 , International Journal for Numerical Methods in Engineering)

   The anisotropic phase‐field dendritic crystal growth model is a highly nonlinear system that couples the anisotropic Allen–Cahn equation and the thermal equation together. Due to the high anisotropy and nonlinear couplings in the system, how to develop an accurate and efficient, especially a fully decoupled scheme, has always been a challenging problem. To solve the challenge, in this article, we construct a novel fully decoupled numerical scheme which is also linear, energy stable, and second‐order time accurate. The key idea to realize the full decoupling structure is to introduce an ordinary differential equation to deal with the nonlinear coupling terms satisfying the so‐called “zero‐energy‐contribution” property. This scheme is very effective and easy to implement since only a few fully decoupled elliptic equations with constant coefficients need to be solved at each time step. We rigorously prove the solvability of each step and the unconditional energy stability, and perform a large number of numerical simulations in 2D and 3D to demonstrate its stability and accuracy numerically.

    

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3. Energy Exchange between Two Sub-systems Coupled with a Nonlinear Elastic Path

   Sen, O.T. ; Singh, R. ( May 2018 , NOVEM 2018 Conference)

   The chief objective of this paper is to explore energy transfer mechanism between the sub-systems that are coupled by a nonlinear elastic path. In the proposed model (via a minimal order, two degree of freedom system), both sub-systems are defined as damped harmonic oscillators with linear springs and dampers. The first sub-system is attached to the ground on one side but connected to the second sub-system on the other side. In addition, linear elastic and dissipative characteristics of both oscillators are assumed to be identical, and a harmonic force excitation is applied only on the mass element of second oscillator. The nonlinear spring (placed in between the two sub-systems) is assumed to exhibit cubic, hardening type nonlinearity. First, the governing equations of the two degree of freedom system with a nonlinear elastic path are obtained. Second, the nonlinear differential equations are solved with a semi-analytical (multi-term harmonic balance) method, and nonlinear frequency responses of the system are calculated for different path coupling cases. As such, the nonlinear path stiffness is gradually increased so that the stiffness ratio of nonlinear element to the linear element is 0.01, 0.05, 0.1, 0.5 and 1.0 while the absolute value of linear spring stiffness is kept intact. In all solutions, it is observed that the frequency response curves at the vicinity of resonant frequencies bend towards higher frequencies as expected due to the hardening effect. However, at moderate or higher levels of path coupling (say 0.1, 0.5 and 1.0), additional branches emerge in the frequency response curves but only at the first resonant frequency. This is due to higher displacement amplitudes at the first resonant frequency as compared to the second one. Even though the oscillators move in-phase around the first natural frequency, high amplitudes increase the contribution of the stored potential energy in the nonlinear spring to the total mechanical energy. The out-of-phase motion around the second natural frequency cannot significantly contribute due to very low motion amplitudes. Finally, the governing equations are numerically solved for the same levels of nonlinearity, and the motion responses of both sub-systems are calculated. Both in-phase and out-of-phase motion responses are successfully shown in numerical solutions, and phase portraits of the system are generated in order to illustrate its nonlinear dynamics. In conclusion, a better understanding of the effect of nonlinear elastic path on two damped harmonic oscillators is gained. 

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4. Second-order stabilized semi-implicit energy stable schemes for bubble assemblies in binary and ternary systems

   https://doi.org/10.3934/dcdsb.2021246  

   Choi, Hyunjung ; Zhao, Yanxiang ( January 2021 , Discrete & Continuous Dynamical Systems - B)

   In this paper, we propose some second-order stabilized semi-implicit methods for solving the Allen-Cahn-Ohta-Kawasaki and the Allen-Cahn-Ohta-Nakazawa equations. In the numerical methods, some nonlocal linear stabilizing terms are introduced and treated implicitly with other linear terms, while other nonlinear and nonlocal terms are treated explicitly. We consider two different forms of such stabilizers and compare the difference regarding the energy stability. The spatial discretization is performed by the Fourier collocation method with FFT-based fast implementations. Numerically, we verify the second order temporal convergence rate of the proposed schemes. In both binary and ternary systems, the coarsening dynamics is visualized as bubble assemblies in hexagonal or square patterns.

    

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5. Discontinuous Galerkin Methods with Time-Operators in Their Numerical Traces for Time-Dependent Electromagnetics

   https://doi.org/10.1515/cmam-2021-0215  

   Cockburn, Bernardo ; Du, Shukai ; Sánchez, Manuel A. ( May 2022 , Computational Methods in Applied Mathematics)

   Abstract We present a new class of discontinuous Galerkin methods for the space discretization of the time-dependent Maxwell equations whose main feature is the use of time derivatives and/or time integrals in the stabilization part of their numerical traces.These numerical traces are chosen in such a way that the resulting semidiscrete schemes exactly conserve a discrete version of the energy.We introduce four model ways of achieving this and show that, when using the mid-point rule to march in time, the fully discrete schemes also conserve the discrete energy.Moreover, we propose a new three-step technique to devise fully discrete schemes of arbitrary order of accuracy which conserve the energy in time.The first step consists in transforming the semidiscrete scheme into a Hamiltonian dynamical system.The second step consists in applying a symplectic time-marching method to this dynamical system in order to guarantee that the resulting fully discrete method conserves the discrete energy in time.The third and last step consists in reversing the above-mentioned transformation to rewrite the fully discrete scheme in terms of the original variables. 

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