2,548 research outputs found
Generalized finite-difference time-domain schemes for solving nonlinear Schrödinger equations
The nonlinear Schrödinger equation (NLSE) is one of the most widely applicable equations in physical science, and characterizes nonlinear dispersive waves, optics, water waves, and the dynamics of molecules. The NLSE satisfies many mathematical conservation laws. Moreover, due to the nonlinearity, the NLSE often requires a numerical solution, which also satisfies the conservation laws. Some of the more popular numerical methods for solving the NLSE include the finite difference, finite element, and spectral methods such as the pseudospectral, split-step with Fourier transform, and integrating factor coupled with a Fourier transform. With regard to the finite difference and finite element methods, higher-order accurate and stable schemes are often required to solve a large-scale linear system. Conversely, spectral methods via Fourier transforms for space discretization coupled with Runge-Kutta methods for time stepping become too complex when applied to multidimensional problems. One of the most prevalent challenges in developing these numerical schemes is that they satisfy the conservation laws.
The objective of this dissertation was to develop a higher-order accurate and simple finite difference scheme for solving the NLSE. First, the wave function was split into real and imaginary components and then substituted into the NLSE to obtain coupled equations. These components were then approximated using higher-order Taylor series expansions in time, where the derivatives in time were replaced by the derivatives in space via the coupled equations. Finally, the derivatives in space were approximated using higher-order accurate finite difference approximations. As such, an explicit and higher order accurate finite difference scheme for solving the NLSE was obtained. This scheme is called the explicit generalized finite-difference time-domain (explicit G-FDTD). For purposes of completeness, an implicit G-FDTD scheme for solving the NLSE was also developed.
In this dissertation, the discrete energy method is employed to prove that both the explicit and implicit G-FDTD scheme satisfy the discrete analogue form of the first conservation law. To verify the accuracy of the numerical solution and the applicability of the schemes, both schemes were tested by simulating bright and dark soliton propagation and collision in one and two dimensions. Compared with other popular existing methods (e.g., pseudospectral, split-step, integrating factor), numerical results showed that the G-FDTD method provides a more accurate solution, particularly when the time step is large. This solution is particularly important during the long-time period simulations. The explicit G-FDTD method proved to be advantageous in that it was simple and fast in computation. Furthermore, the G-FDTD showed that the solution propagates through the boundary with analytical solution continuation
Fourth order real space solver for the time-dependent Schr\"odinger equation with singular Coulomb potential
We present a novel numerical method and algorithm for the solution of the 3D
axially symmetric time-dependent Schr\"odinger equation in cylindrical
coordinates, involving singular Coulomb potential terms besides a smooth
time-dependent potential. We use fourth order finite difference real space
discretization, with special formulae for the arising Neumann and Robin
boundary conditions along the symmetry axis. Our propagation algorithm is based
on merging the method of the split-operator approximation of the exponential
operator with the implicit equations of second order cylindrical 2D
Crank-Nicolson scheme. We call this method hybrid splitting scheme because it
inherits both the speed of the split step finite difference schemes and the
robustness of the full Crank-Nicolson scheme. Based on a thorough error
analysis, we verified both the fourth order accuracy of the spatial
discretization in the optimal spatial step size range, and the fourth order
scaling with the time step in the case of proper high order expressions of the
split-operator. We demonstrate the performance and high accuracy of our hybrid
splitting scheme by simulating optical tunneling from a hydrogen atom due to a
few-cycle laser pulse with linear polarization
ABC Method and Fractional Momentum Layer for the FDTD Method to Solve the Schrödinger Equation on Unbounded Domains
The finitedifference timedomain (FDTD) method and its generalized variant (GFDTD) are efficient numerical tools for solving the linear and nonlinear Schrödinger equations because not only are they explicit, allowing parallelization, but they also provide highorder accuracy with relatively inexpensive computational costs. In addition, the GFDTD method has a relaxed stability condition when compared to the original FDTD method. It is important to note that the existing simulations of the GFDTD scheme employed analytical solutions to obtain function values at the points along the boundary; however, in simulations for which the analytical solution is unknown, theoretical approximations for values at points along the boundary are desperately needed. Hence, the objective of this dissertation research is to develop absorbing boundary conditions (ABCs) so that the GFDTD method can be used to solve the nonlinear Schrödinger equation when the analytical solution is unknown.
To create the ABCs for the nonlinear Schrödinger equation, we initially determine the associated EngquistMajda oneway wave equations and then proceed to develop a finite difference scheme for them. These ABCs are made to be adaptive using a windowed Fourier transform to estimate a value of the wavenumber of the carrier wave. These ABCs were tested using the nonlinear Schrödinger equation for 1D and 2D soliton propagation as well as Gaussian packet collision and dipole radiation. Results show that these ABCs perform well, but they have three key limitations. First, there are inherent reflections at the interface of the interior and boundary domains due to the different schemes used the two regions; second, to use the ABCs, one needs to estimate a value for the carrier wavenumber and poor estimates can cause even more reflection at the interface; and finally, the ABCs require different schemes in different regions of the boundary, and this domain decomposition makes the ABCs tedious both to develop and to implement.
To address these limitations for the FDTD method, we employ the fractionalorder derivative concept to unify the Schrödinger equation with its oneway wave equation over an interval where the fractional order is allowed to vary. Through careful construction of a variableorder fractional momentum operator, outgoing waves may enter the fractionalorder region with little to no reflection and, inside this region, any reflected portions of the wave will decay exponentially with time. The fractional momentum operator is then used to create a fractionalorder FDTD scheme. Importantly, this single scheme can be used for the entire computational domain, and the scheme smooths the abrupt transition between the FDTD method and the ABCs. Furthermore, the fractional FDTD scheme relaxes the precision needed for the estimated carrier wavenumber. This fractional FDTD scheme is tested for both the linear and nonlinear Schrödinger equations. Example cases include a 1D Gaussian packet scattering off of a potential, a 1D soliton propagating to the right, as well as 2D soliton propagation, and the collision of Gaussian packets. Results show that the fractional FDTD method outperforms the FDTD method with ABCs
Approximated Lax Pairs for the Reduced Order Integration of Nonlinear Evolution Equations
A reduced-order model algorithm, called ALP, is proposed to solve nonlinear
evolution partial differential equations. It is based on approximations of
generalized Lax pairs. Contrary to other reduced-order methods, like Proper
Orthogonal Decomposition, the basis on which the solution is searched for
evolves in time according to a dynamics specific to the problem. It is
therefore well-suited to solving problems with progressive front or wave
propagation. Another difference with other reduced-order methods is that it is
not based on an off-line / on-line strategy. Numerical examples are shown for
the linear advection, KdV and FKPP equations, in one and two dimensions
Exact equations for smoothed Wigner transforms and homogenization of wave propagation
The Wigner Transform (WT) has been extensively used in the formulation of
phase-space models for a variety of wave propagation problems including
high-frequency limits, nonlinear and random waves. It is well known that the WT
features counterintuitive 'interference terms', which often make computation
impractical. In this connection, we propose the use of the smoothed Wigner
Transform (SWT), and derive new, exact equations for it, covering a broad class
of wave propagation problems. Equations for spectrograms are included as a
special case. The 'taming' of the interference terms by the SWT is illustrated,
and an asymptotic model for the Schroedinger equation is constructed and
numerically verified.Comment: 16 pages, 8 figure
Numerical algorithms for Schrödinger equation with artificial boundary conditions
We consider a one-dimensional linear Schrödinger problem defined on an infinite domain and approximated by the Crank-Nicolson type finite difference scheme. To solve this problem numerically we restrict the computational domain by introducing the reflective, absorbing or transparent artificial boundary conditions. We investigate the conservativity of the discrete scheme with respect to the mass and energy of the solution. Results of computational experiments are presented and the efficiency of different artificial boundary conditions is discussed
Numerical methods for generalized nonlinear Schrödinger equations
We present and analyze different splitting algorithms for numerical solution of the both classical and generalized nonlinear Schr"odinger equations describing propagation of wave packets with special emphasis on applications to nonlinear fiber-optics. The considered generalizations take into account the higher-order corrections of the linear differential dispersion operator as well as the saturation of nonlinearity and the self-steepening of the field envelope function. For stabilization of the pseudo-spectral splitting schemes for generalized Schr"odinger equations a regularization based on the approximation of the derivatives by the low number of Fourier modes is proposed. To illustrate the theoretically predicted performance of these schemes several numerical experiments have been done
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