A new adaptive cell average spectral element method (SEM) is proposed to solve the time-dependent Wigner equation for transport in quantum devices. The proposed cell average SEM allows adaptive non-uniform meshes in phase spaces to reduce the high-dimensional computational cost of Wigner functions while preserving exactly the mass conservation for the numerical solutions. The key feature of the proposed method is an analytical relation between the cell averages of the Wigner function in the k-space (local electron density for finite range velocity) and the point values of the distribution, resulting in fast transforms between the local electron density and local fluxes of the discretized Wigner equation via the fast sine and cosine transforms. Numerical results with the proposed method are provided to demonstrate its high accuracy, conservation, convergence and a reduction of the cost using adaptive meshes.
This paper presents a review of the current state-of-the-art of numerical methods for nonlinear Dirac (NLD) equation. Several methods are extendedly proposed for the (1+1)-dimensional NLD equation with the scalar and vector self-interaction and analyzed in the way of the accuracy and the time reversibility as well as the conservation of the discrete charge, energy and linear momentum. Those methods are the Crank-Nicolson (CN) schemes, the linearized CN schemes, the odd-even hopscotch scheme, the leapfrog scheme, a semi-implicit finite difference scheme, and the exponential operator splitting (OS) schemes. The nonlinear subproblems resulted from the OS schemes are analytically solved by fully exploiting the local conservation laws of the NLD equation. The effectiveness of the various numerical methods, with special focus on the error growth and the computational cost, is illustrated on two numerical experiments, compared to two high-order accurate Runge-Kutta discontinuous Galerkin methods. Theoretical and numerical comparisons show that the high-order accurate OS schemes may compete well with other numerical schemes discussed here in terms of the accuracy and the efficiency. A fourth-order accurate OS scheme is further applied to investigating the interaction dynamics of the NLD solitary waves under the scalar and vector self-interaction. The results show that the interaction dynamics of two NLD solitary waves depend on the exponent power of the self-interaction in the NLD equation; collapse happens after collision of two equal one-humped NLD solitary waves under the cubic vector self-interaction in contrast to no collapse scattering for corresponding quadric case.Comment: 39 pages, 13 figure
We consider the nonlinear Dirac equation in 1 + 1 dimension with scalar-scalar self interactionand with mass m. Using the exact analytic form for rest frame solitary waves of the form (x,t) = ψ(x)e −iωt for arbitrary κ, we discuss the validity of various approaches to understanding stability that were successful for the nonlinear Schrödinger equation. In particular we study the validity of a version of Derrick's theorem and the criterion of Bogolubsky as well as the Vakhitov-Kolokolov criterion, and find that these criteria yield inconsistent results. Therefore, we study the stability by numerical simulations using a recently developed fourth-order operator splitting integration method. For different ranges of κ we map out the stability regimes in ω. We find that all stable nonlinear Dirac solitary waves have a one-hump profile, but not all one-hump waves are stable, while all waves with two humps are unstable. We also find that the time t c , it takes for the instability to set in, is an exponentially increasing function of ω and t c decreases monotonically with increasing κ.
To set up the general framework for relativistic explicitly correlated wave function methods, the electron-electron coalescence conditions are derived for the wave functions of the Dirac-Coulomb (DC), Dirac-Coulomb-Gaunt (DCG), Dirac-Coulomb-Breit (DCB), modified Dirac-Coulomb (MDC), and zeroth-order regularly approximated (ZORA) Hamiltonians. The manipulations make full use of the internal symmetries of the reduced two-electron Hamiltonians such that the asymptotic behaviors of the wave functions emerge naturally. The results show that, at the coalescence point of two electrons, the wave functions of the DCG Hamiltonian are regular, while those of the DC and DCB Hamiltonians have weak singularities of the type r(12)(ν) with ν being negative and of O(α(2)). The behaviors of the MDC wave functions are related to the original ones in a simple manner, while the spin-free counterparts are somewhat different due to the complicated electron-electron interaction. The behaviors of the ZORA wave functions depend on the chosen potential in the kinetic energy operator. In the case of the nuclear attraction, the behaviors of the ZORA wave functions are very similar to those of the nonrelativistic ones, just with an additional correction of O(α(2)) to the nonrelativistic cusp condition. However, if the Coulomb interaction is also included, the ZORA wave functions become close to the large-large components of the DC wave functions. Note that such asymptotic expansions of the relativistic wave functions are only valid within an extremely small convergence radius R(c) of O(α(2)). Beyond this radius, the behaviors of the relativistic wave functions are still dominated by the nonrelativistic limit, as can be seen in terms of direct perturbation theory (DPT) of relativity. However, as the two limits α → 0 and r(12) → 0 do not commute, DPT is doomed to fail due to incorrect descriptions of the small-small component Ψ(SS) of the DC wave function for r(12) < R(c). Another deduction from the possible divergence of Ψ(SS) at r(12) = R(c) is that the DC Hamiltonian has no bound electronic states, although the last word cannot be said. These findings enrich our understandings of relativistic wave functions. On the practical side, it is shown that, under the no-pair approximation, relativistic explicitly correlated wave function methods can be made completely parallel to the nonrelativistic counterparts, as demonstrated explicitly for MP2-F12. Yet, this can only be achieved by using an extended no-pair projector.
We analyze the time reversible Born-Oppenheimer molecular dynamics (TRBOMD) scheme, which preserves the time reversibility of the Born-Oppenheimer molecular dynamics even with non-convergent self-consistent field iteration. In the linear response regime, we derive the stability condition, as well as the accuracy of TRBOMD for computing physical properties, such as the phonon frequency obtained from the molecular dynamics simulation. We connect and compare TRBOMD with Car-Parrinello molecular dynamics in terms of accuracy and stability. We further discuss the accuracy of TRBOMD beyond the linear response regime for non-equilibrium dynamics of nuclei. Our results are demonstrated through numerical experiments using a simplified one-dimensional model for Kohn-Sham density functional theory.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.