A Generalized FDTD Method with Absorbing Boundary Condition for Solving a Time-Dependent Linear Schrodinger Equation

The Finite-Difference Time-Domain (FDTD) method is a well-known technique for the analysis of quantum devices. It solves a discretized Schrodinger equation in an iterative process. However, the method provides only a second-order accurate numerical solution and requires that the spatial grid size and time step should satisfy a very restricted condition in order to prevent the numerical solution from diverging. In this article, we present a generalized FDTD method with absorbing boundary condition for solving the one-dimensional (1D) time-dependent Schr?dinger equation and obtain a more relaxed condition for stability. The generalized FDTD scheme is tested by simulating a particle moving in free space and then hitting an energy potential. Numerical results coincide with those obtained based on the theoretical analysis.

Numerical Algorithm with Fourth-Order Spatial Accuracy for Solving the Time-Fractional Dual-Phase-Lagging Nanoscale Heat Conduction Equation

Nanoscale heat transfer cannot be described by the classical Fourier law due to the very small dimension,and therefore,analyzing heat transfer in nanoscale is of crucial importance for the design and operation of nano-devices and the optimization of thermal processing of nano-materials.Recently,time-fractional dualphase-lagging(DPL)equations with temperature jump boundary conditions have showed promising for analyzing the heat conduction in nanoscale.This article proposes a numerical algorithm with high spatial accuracy for solving the timefractional dual-phase-lagging nano-heat conduction equation with temperature jump boundary conditions.To this end,we first develop a fourth-order accurate and unconditionally stable compact finite difference scheme for solving this time-fractional DPL model.We then present a fast numerical solver based on the divide-and-conquer strategy for the obtained finite difference scheme in order to reduce the huge computational work and storage.Finally,the algorithm is tested by two examples to verify the accuracy of the scheme and computational speed.And we apply the numerical algorithm for predicting the temperature rise in a nano-scale silicon thin film.Numerical results confirm that the present difference scheme provides min{2−α,2−β}order accuracy in time and fourth-order accuracy in space,which coincides with the theoretical analysis.Results indicate that the mentioned time-fractional DPL model could be a tool for investigating the thermal analysis in a simple nanoscale semiconductor silicon device by choosing the suitable fractional order of Caputo derivative and the parameters in the model.

A New Artificial Neural Network Method for Solving Schrodinger Equations on Unbounded Domains

The simulation for particle or soliton propagation based on linear or nonlinear Schrodinger equations on unbounded domains requires the computational domain to be bounded,and therefore,a special boundary treatment such as an absorbing boundary condition(ABC)or a perfectly matched layer(PML)is needed so that the reflections of outgoing waves at the boundary can be minimized in order to prevent the destruction of the simulation.This article presents a new artificial neural network(ANN)method for solving linear and nonlinear Schrodinger equations on unbounded domains.In particular,this method randomly selects training points only from the bounded computational space-time domain,and the loss function involves only the initial condition and the Schrodinger equation itself in the computational domainwithout any boundary conditions.Moreover,unlike standard ANNmethods that calculate gradients using expensive automatic differentiation,this method uses accurate finitedifference approximations for the physical gradients in the Schrodinger equation.In addition,a Metropolis-Hastings algorithm is implemented for preferentially selecting regions of high loss in the computational domain allowing for the use of fewer training points in each batch.As such,the present training method uses fewer training points and less computation time for convergence of the loss function as compared with the standard ANN methods.This new ANN method is illustrated using three examples.