An Optimized Perfectly Matched Layer for the Schrödinger Equation

Nissen, Anna; Kreiss, Gunilla

doi:10.4208/cicp.010909.010410a

Cited by 34 publications

(36 citation statements)

References 9 publications

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“…This discretisation, as any other discretisation of the modal PML [25], has not been proven to be stable, but we do not experience any signs of numerical instabilities in our computations.…”

Section: Absorbing Boundary Conditionsmentioning

confidence: 93%

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Coupling of Gaussian Beam and Finite Difference Solvers for Semiclassical Schrödinger Equations

Kieri

Kreiss

Runborg

2015

Adv. Appl. Math. Mech.

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In the semiclassical regime, solutions to the time-dependent Schrödinger equation are highly oscillatory. The number of grid points required for resolving the oscillations may become very large even for simple model problems, making solution on a grid, e.g., using a finite difference method, intractable. Asymptotic methods like Gaussian beams can resolve the oscillations with little effort and yield good approximations when the molecules are heavy and the potential is smooth. However, when the potential has variations on a small length-scale, quantum phenomena become important. Then asymptotic methods are less accurate. The two classes of methods perform well in different parameter regimes. This opens for hybrid methods, using Gaussian beams where we can and finite differences where we have to. We propose a new method for treating the coupling between the finite difference method and Gaussian beams. The new method reduces the needed amount of overlap regions considerably compared to previous methods, which improves the efficiency. We apply the method to scattering problems in one and two dimensions.

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Section: Absorbing Boundary Conditionsmentioning

confidence: 93%

“…We use the modal PML presented in [25], which in one dimension is implemented by modifying the kinetic part of the Hamiltonian to…”

Section: Absorbing Boundary Conditionsmentioning

confidence: 99%

Coupling of Gaussian Beam and Finite Difference Solvers for Semiclassical Schrödinger Equations

Kieri

Kreiss

Runborg

2015

Adv. Appl. Math. Mech.

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“…The idea is to enlarge the computational domain by an artificial damping layer of finite width, where a modified equation has to be solved. The method has been applied to the Schrödinger equation in [11,26].…”

Section: Introductionmentioning

confidence: 99%

Time-dependent simulations of quantum waveguides using a time-splitting spectral method

Jüngel¹,

Mennemann²

2010

Mathematics and Computers in Simulation

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The electron flow through quantum waveguides is modeled by the time-dependent Schrö-dinger equation with absorbing boundary conditions, which are realized by a negative imaginary potential. The Schrödinger equation is discretized by a time-splitting spectral method, and the quantum waveguides are fed by a mono-energetic incoming plane wave pulse. The resulting algorithm is extremely efficient due to the Fast Fourier Transform implementation of the spectral scheme. Numerical convergence rates for a one-dimensional scattering problem are calculated. The transmission rates of a two-dimensional T-stub quantum waveguide and a single-branch coupler are numerically computed. Moreover, the transient behavior of a three-dimensional T-stub waveguide is simulated.

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“…The discretization of the Hamiltonian is obtained via a 7-point stencil. The self-energy matrix Σ r is assumed to be represented via a similar 7-point stencil (for example, with a crude diagonal approximation or with a PML-like approximation [22]). The resulting matrix A is of dimension N x × N z × N y , where the y-direction is the transport direction.…”

Section: Operation Count Analysis When Simulating 3d Brick-like Devicesmentioning

confidence: 99%

Nested dissection solver for transport in 3D nano-electronic devices

et al. 2016

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The Hierarchical Schur Complement method (HSC), and the HSC-extension, have significantly accelerated the evaluation of the retarded Green's function, particularly the lesser Green's function, for two-dimensional nanoscale devices. In this work, the HSC-extension is applied to determine the solution of non-equilibrium Green's functions (NEGF) on three-dimensional nanoscale devices. The operation count for the HSC-extension is analyzed for a cuboid device. When a cubic device is discretized with N × N × N grid points, the state-of-the-art Recursive Green Function (RGF) algorithm takes O(N 7 ) operations, whereas the HSC-extension only requires O(N 6 ) operations. Operation counts and runtimes are also studied for three-dimensional nanoscale devices of practical interest: a graphene-boron nitride-graphene multilayer system, a silicon nanowire, and a DNA molecule. The numerical experiments indicate that the cost for the HSCextension is proportional to the solution of one linear system (or one LU-factorization) and that the runtime speedups over RGF exceed three orders of magnitude when simulating realistic devices, such as a graphene-boron nitridegraphene multilayer system with 40,000 atoms.

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An Optimized Perfectly Matched Layer for the Schrödinger Equation

Cited by 34 publications

References 9 publications

Coupling of Gaussian Beam and Finite Difference Solvers for Semiclassical Schrödinger Equations

Coupling of Gaussian Beam and Finite Difference Solvers for Semiclassical Schrödinger Equations

Time-dependent simulations of quantum waveguides using a time-splitting spectral method

Nested dissection solver for transport in 3D nano-electronic devices

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