In this paper we study the numerical method and the convergence for solving the time-dependent Maxwell-Schrödinger equations under the Lorentz gauge. An alternating Crank-Nicolson finite element method for solving the problem is presented and the optimal error estimate for the numerical algorithm is obtained by a mathematical inductive method. Numerical examples are then carried out to confirm the theoretical results.
In this paper, we consider the initial-boundary value problem for the time-dependent Maxwell--Schrödinger equations in the Coulomb gauge. We first prove the global existence of weak solutions to the equations. Next we propose an energy-conserving fully discrete finite element scheme for the system and prove the existence and uniqueness of solutions to the discrete system. The optimal error estimates for the numerical scheme without any time-step restrictions are then derived. Numerical results are provided to support our theoretical analysis.
In this paper we consider the initial-boundary value problem for the time-dependent Maxwell-Schrödinger equations, which arises in the interaction between the matter and the electromagnetic field for the semiconductor quantum devices. A Crank-Nicolson finite element method for solving the problem is presented. The optimal energy-norm error estimates for the numerical algorithm without any time-step restrictions are derived. Numerical tests are then carried out to confirm the theoretical results.
This paper discusses the multiscale approach and the convergence of the time-dependent Maxwell-Schrödinger system with rapidly oscillating discontinuous coefficients arising from the modeling of a heterogeneous nanostructure with a periodic microstructure. The homogenization method and the multiscale asymptotic method for the nonlinear coupled equations are presented. The efficient numerical algorithms based on the above methods are proposed. Numerical simulations are then carried out to validate the method presented in this paper.
Understanding and prediction of the chemical reactions are fundamental demanding in the study of many complex chemical systems. Reactive molecular dynamics (MD) simulation has been widely used for this purpose as it can offer atomic details and can help us better interpret chemical reaction mechanisms. In this study, two reference datasets were constructed and corresponding neural network (NN) potentials were trained based on them. For given large-scale reaction systems, the NN potentials can predict the potential energy and atomic forces of DFT precision, while it is orders of magnitude faster than the conventional DFT calculation. With these two models, reactive MD simulations were performed to explore the combustion mechanisms of hydrogen and methane. Benefit from the high efficiency of the NN model, nanosecond MD trajectories for large-scale systems containing hundreds of atoms were produced and detailed combustion mechanism was obtained. Through further development, the algorithms in this study can be used to explore and discovery reaction mechanisms of many complex reaction systems, such as combustion, synthesis, and heterogeneous catalysis without any predefined reaction coordinates and elementary reaction steps.