arXiv subjects

Jingrun Chen

Publications and source records attributed to Jingrun Chen.

66 records · Page 4Linked to original sources

arXiv2019

Spin waves in ferromagnetic thin films

A spin wave is the disturbance of intrinsic spin order in magnetic materials. In this paper, a spin wave in the Landau-Lifshitz-Gilbert equation is obtained based on the assumption that the spin wave maintains its shape while it propagates at a constant velocity. Our main findings include: (1) in the absence of Gilbert damping, the spin wave propagates at a constant velocity with the increment proportional to the strength of the magnetic field; (2) in the absence of magnetic field, at a given time the spin wave converges exponentially fast to its initial profile as the damping parameter goes to zero and in the long time the relaxation dynamics of the spin wave converges exponentially fast to the easy-axis direction with the exponent proportional to the damping parameter; (3) in the presence of both Gilbert damping and magnetic field, the spin wave converges to the easy-axis direction exponentially fast at a small timescale while propagates at a constant velocity beyond that. These provides a comprehensive understanding of spin waves in ferromagnetic materials.

cond-mat.mes-hall↗

arXiv2019

Efficient multiscale methods for the semiclassical Schrödinger equation with time-dependent potentials

The semiclassical Schrödinger equation with time-dependent potentials is an important model to study electron dynamics under external controls in the mean-field picture. In this paper, we propose two multiscale finite element methods to solve this problem. In the offline stage, for the first approach, the localized multiscale basis functions are constructed using sparse compression of the Hamiltonian operator at the initial time; for the latter, basis functions are further enriched using a greedy algorithm for the sparse compression of the Hamiltonian operator at later times. In the online stage, the Schrödinger equation is approximated by these localized multiscale basis in space and is solved by the Crank-Nicolson method in time. These multiscale basis have compact supports in space, leading to the sparsity of stiffness matrix, and thus the computational complexity of these two methods in the online stage is comparable to that of the standard finite element method. However, the spatial mesh size in multiscale finite element methods is $ H=\mathcal{O}(ε) $, while $H=\mathcal{O}(ε^{3/2})$ in the standard finite element method, where $ε$ is the semiclassical parameter. By a number of numerical examples in 1D and 2D, for approximately the same number of basis, we show that the approximation error of the multiscale finite element method is at least two orders of magnitude smaller than that of the standard finite element method, and the enrichment further reduces the error by another one order of magnitude.

cs.CE↗

arXiv2019

Two improved Gauss-Seidel projection methods for Landau-Lifshitz-Gilbert equation

In this paper, we present two improved Gauss-Seidel projection methods with unconditional stability. The first method updates the gyromagnetic term and the damping term simultaneously and follows by a projection step. The second method introduces two sets of approximate solutions, where we update the gyromagnetic term and the damping term simultaneously for one set of approximate solutions and apply the projection step to the other set of approximate solutions in an alternating manner. Compared to the original Gauss-Seidel projection method which has to solve heat equations $7$ times at each time step, the improved methods solve heat equations $5$ times and $3$ times, respectively. First-order accuracy in time and second-order accuracy in space are verified by examples in both 1D and 3D. In addition, unconditional stability with respect to both the grid size and the damping parameter is confirmed numerically. Application of both methods to a realistic material is also presented with hysteresis loops and magnetization profiles. Compared with the original method, the recorded running times suggest that savings of both methods are about $2/7$ and $4/7$ for the same accuracy requirement, respectively.

math.NA↗

arXiv2019

Convergence Analysis of A Second-order Semi-implicit Projection Method for Landau-Lifshitz Equation

The numerical approximation for the Landau-Lifshitz equation, the dynamics of magnetization in a ferromagnetic material, is taken into consideration. This highly nonlinear equation, with a non-convex constraint, has several equivalent forms, and involves solving an auxiliary problem in the infinite domain. All these features have posed interesting challenges in developing numerical methods. In this paper, we first present a fully discrete semi-implicit method for solving the Landau-Lifshitz equation based on the second-order backward differentiation formula and the one-sided extrapolation (using previous time-step numerical values). A projection step is further used to preserve the length of the magnetization. Subsequently, we provide a rigorous convergence analysis for the fully discrete numerical solution, with second-order accuracy in both time and space, provided that the spatial step-size is the same order as the temporal step-size. And also, the unique solvability of the numerical solution is theoretically justified, which turns out to be the first such result for the micromagnetics model. All these theoretical properties are verified by numerical examples in both one- and three- dimensional spaces.

math.AP↗

arXiv2019

Second-order semi-implicit projection methods for micromagnetics simulations

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.

physics.comp-ph↗

arXiv2019

A multiscale reduced basis method for Schrödinger equation with multiscale and random potentials

The semiclassical Schrödinger equation with multiscale and random potentials often appears when studying electron dynamics in heterogeneous quantum systems. As time evolves, the wavefunction develops high-frequency oscillations in both the physical space and the random space, which poses severe challenges for numerical methods. In this paper, we propose a multiscale reduced basis method, where we construct multiscale reduced basis functions using an optimization method and the proper orthogonal decomposition method in the physical space and employ the quasi-Monte Carlo method in the random space. Our method is verified to be efficient: the spatial gridsize is only proportional to the semiclassical parameter and the number of samples in the random space is inversely proportional to the same parameter. Several theoretical aspects of the proposed method, including how to determine the number of samples in the construction of multiscale reduced basis and convergence analysis, are studied with numerical justification. In addition, we investigate the Anderson localization phenomena for Schrödinger equation with correlated random potentials in both 1D and 2D.

math.NA↗

arXiv2019

A multiscale finite element method for the Schrödinger equation with multiscale potentials

In recent years, an increasing attention has been paid to quantum heterostructures with tailored functionalities, such as heterojunctions and quantum matematerials, in which quantum dynamics of electrons can be described by the Schrödinger equation with multiscale potentials. The model, however, cannot be solved by asymptoics-based approaches where an additive form of different scales in the potential term is required to construct the prescribed approximate solutions. In this paper, we propose a multiscale finite element method to solve this problem in the semiclassical regime. The localized multiscale basis are constructed using sparse compression of the Hamiltonian operator, and thus are "blind" to the specific form of the potential term. After an one-shot eigendecomposition, we solve the resulting system of ordinary differential equations explicitly for the time evolution. In our approach, the spatial mesh size is $ H=\mathcal{O}(ε) $ where $ε$ is the semiclassical parameter and the time stepsize $ k$ is independent of $ε$. Numerical examples in one dimension with a periodic potential, a multiplicative two-scale potential, and a layered potential, and in two dimension with an additive two-scale potential and a checkboard potential are tested to demonstrate the robustness and efficiency of the proposed method. Moreover, first-order and second-order rates of convergence are observed in $H^1$ and $L^2$ norms, respectively.

math.NA↗

arXiv2017

Estimation of exciton diffusion lengths of organic semiconductors in random domains

Exciton diffusion length plays a vital role in the function of opto-electronic devices. Oftentimes, the domain occupied by an organic semiconductor is subject to surface measurement error. In many experiments, photoluminescence over the domain is measured and used as the observation data to estimate this length parameter in an inverse manner based on the least square method. However, the result is sometimes found to be sensitive to the surface geometry of the domain. In this paper, we employ a random function representation for the uncertain surface of the domain. After non-dimensionalization, the forward model becomes a diffusion-type equation over the domain whose geometric boundary is subject to small random perturbations. We propose an asymptotic-based method as an approximate forward solver whose accuracy is justified both theoretically and numerically. It only requires solving several deterministic problems over a fixed domain. Therefore, for the same accuracy requirements we tested here, the running time of our approach is more than one order of magnitude smaller than that of directly solving the original stochastic boundary-value problem by the stochastic collocation method. In addition, from numerical results, we find that the correlation length of randomness is important to determine whether a 1D reduced model is a good surrogate for the 2D model.

math.NA↗

arXiv2017

Two-parameter asymptotic expansions for elliptic equations with small geometric perturbation and high contrast ratio

We consider the asymptotic solutions of an interface problem corresponding to an elliptic partial differential equation with Dirich- let boundary condition and transmission condition, subject to the small geometric perturbation and the high contrast ratio of the conductivity. We consider two types of perturbations: the first corresponds to a thin layer coating a fixed bounded domain and the second is the per perturbation of the interface. As the perturbation size tends to zero and the ratio of the conductivities in two subdomains tends to zero, the two-parameter asymptotic expansions on the fixed reference domain are derived to any order after the single parameter expansions are solved be- forehand. Our main tool is the asymptotic analysis based on the Taylor expansions for the properly extended solutions on fixed domains. The Neumann boundary condition and Robin boundary condition arise in two-parameter expansions, depending on the relation of the geometric perturbation size and the contrast ratio.

math.AP↗

arXiv2015

An atomistic/continuum coupling method using enriched bases

A common observation from an atomistic to continuum coupling method is that the error is often generated and concentrated near the interface, where the two models are combined. In this paper, a new method is proposed to suppress the error at the interface, and as a consequence, the overall accuracy is improved. The method is motivated by formulating the molecular mechanics model as a two-stage minimization problem. In particular, it is demonstrated that the error at the interface can be considerably reduced when new basis functions are introduced in a Galerkin projection formalism. The improvement of the accuracy is illustrated by two examples. Further, the comparison to some quasicontinuum-type methods is provided.

math.NA↗

arXiv2015

Analysis of the divide-and-conquer method for electronic structure calculations

We study the accuracy of the divide-and-conquer method for electronic structure calculations. The analysis is conducted for a prototypical subdomain problem in the method. We prove that the pointwise difference between electron densities of the global system and the subsystem decays exponentially as a function of the distance away from the boundary of the subsystem, under the gap assumption of both the global system and the subsystem. We show that gap assumption is crucial for the accuracy of the divide-and-conquer method by numerical examples. In particular, we show examples with the loss of accuracy when the gap assumption of the subsystem is invalid.

math.NA↗

arXiv2012

Ghost Force Influence of a Quasicontinuum Method in Two Dimension

We derive an analytical expression for the solution of a two-dimensional quasicontinuum method with a planar interface. The expression is used to prove that the ghost force may lead to a finite size error for the gradient of the solution. We estimate the width of the interfacial layer induced by the ghost force is of $\co(\sqrt{\eps}\,)$ with $\eps$ the equilibrium bond length, which is much wider than that of the one-dimensional problem.

math.NA↗