Nonuniform Finite Difference Method Research Articles

Numerical methods for scalar field dark energy in tabletop experiments and Lunar Laser Ranging

Numerous tabletop experiments have been dedicated to exploring the manifestations of screened scalar field dark energy, such as symmetron or chameleon fields. Precise theoretical predictions require simulating field configurations within the respective experiments. This paper focuses onto the less-explored environment-dependent dilaton field, which emerges in the strong coupling limit of string theory. Due to its exponential self-coupling, this field can exhibit significantly steeper slopes compared to symmetron and chameleon fields, and the equations of motion can be challenging to solve with standard machine precision. We present the first exact solution for the geometry of a vacuum region between two infinitely extended parallel plates. This solution serves as a benchmark for testing the accuracy of numerical solvers. By reparametrizing the model and transforming the equations of motion, we show how to make the model computable across the entire experimentally accessible parameter space. To simulate the dilaton field in one- and two-mirror geometries, as well as spherical configurations, we introduce a non-uniform finite difference method. Additionally, we provide an algorithm for solving the stationary Schrödinger equation for a fermion in one dimension in the presence of a dilaton field. The algorithms developed here are not limited to the dilaton field, but can be applied to similar scalar-tensor theories as well. We demonstrate such applications at hand of the chameleon and symmetron field. Our computational tools have practical applications in a variety of experimental contexts, including gravity resonance spectroscopy (q Bounce), Lunar Laser Ranging (LLR), and the upcoming Casimir and Non-Newtonian Force Experiment (cannex). A Mathematica implementation of all algorithms is provided.

Highly efficient nonuniform finite difference method for three-dimensional electrically stimulated liquid crystal photonic devices

Liquid crystal (LC) photonic devices have attracted intensive attention in recent decades, due to the merits of tunability, cost-effectiveness, and high efficiency. However, the precise and efficient simulation of large-scale three-dimensional electrically stimulated LC photonic devices remains challenging and resource consuming. Here we report a straightforward nonuniform finite difference method (NFDM) for efficiently simulating large-scale LC photonic devices by employing a spatially nonuniform mesh grid. We show that the NFDM can be further accelerated by approximately 504 times by using the improved successive over-relaxation method (by 12 times), the symmetric boundary (by 4 times), the momentum gradient descent algorithm (by 3.5 times), and the multigrid (by 3 times). We experimentally fabricated the large-scale electrically stimulated LC photonic device, and the measured results demonstrate the effectiveness and validity of the proposed NFDM. The NFDM allocates more grids to the core area with steep electric field gradient, thus reducing the distortion of electric field and the truncation error of calculation, rendering it more precise than the finite element method and traditional finite difference method with similar computing resources. This study demonstrates an efficient and highly reliable method to simulate the large-scale electrically stimulated LC photonic device, and paves the way for customizing a large-scale LC photonic device with designable functionalities.

An Algorithm for Fast Simulation of CW Illuminator

The continuous wave (CW) illuminator is aimed to simulate high-altitude nuclear electromagnetic pulse. In this article, a simulation algorithm is proposed covering the method of moments and finite difference method (FDM), with the high-speed simulation of the CW illuminator with arbitrary ellipse size, arbitrary loading, and arbitrary polarization direction. Based on the surface electric field equation of the CW illuminator, the mathematical model is established with the vector potential equation and the boundary conditions. In this model, the pulse function and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\delta$</tex-math></inline-formula> function are set as the basis function and the weight function, respectively, for the establishment of the current equation, with the current distribution determined by the simplification equation with image theory and nonuniform FDM. Finally, as the field distribution is determined by the superposition theorem. The simulation algorithm is verified by the self-built CW illuminator.

Nonuniform Finite Difference Scheme for the Three-Dimensional Time-Fractional Black–Scholes Equation

In this study, we present an accurate and efficient nonuniform finite difference method for the three-dimensional (3D) time-fractional Black–Scholes (BS) equation. The operator splitting scheme is used to efficiently solve the 3D time-fractional BS equation. We use a nonuniform grid for pricing 3D options. We compute the three-asset cash-or-nothing European call option and investigate the effects of the fractional-orderαin the time-fractional BS model. Numerical experiments demonstrate the efficiency and fastness of the proposed scheme.

An inverse problem for finding the lowest term of a heat equation with Wentzell–Neumann boundary condition

ABSTRACTThe Fourier series analysis of the inverse problem of finding the coefficient of lowest term in the heat equation with a non-local Wentzell–Neumann boundary and integral overdetermination conditions is presented. Under some regularity, consistency and orthogonality conditions on the data and additional conditions on the sign of the Fourier coefficients of the initial data and known part of source term, the existence and uniqueness of the classical solution are shown by using the generalized Fourier method. Numerical solutions of the inverse problem are given both on a uniform grid and on a non-uniform grid with uniform finite difference method combined with the composite trapezoidal rule and with non-uniform finite difference method combined with Gauss–Lobatto quadrature, respectively. Two numerical examples (one is smooth while the other is non-smooth) will be provided to investigate the efficiency and the stability of the methods with respect to the coefficient of diffusive transport to the boundary.

Nonuniform Crank‐Nicolson scheme for solving the stochastic Kawarada equation via arbitrary grids

This article studies a nonuniform finite difference method for solving the degenerate Kawarada quenching‐combustion equation with a vibrant stochastic source. Arbitrary grids are introduced in both space and time via adaptive principals to accommodate the uncertainty and singularities involved. It is shown that, under proper constraints on mesh step sizes, the positivity, monotonicity of the solution, and numerical stability of the scheme developed are well preserved. Numerical experiments are given to illustrate our conclusions. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1305–1328, 2017

Accurate and Efficient Computations of the Greeks for Options Near Expiry Using the Black-Scholes Equations

We investigate the accurate computations for the Greeks using the numerical solutions of the Black-Scholes partial differential equation. In particular, we study the behaviors of the Greeks close to the maturity time and in the neighborhood around the strike price. The Black-Scholes equation is discretized using a nonuniform finite difference method. We propose a new adaptive time-stepping algorithm based on local truncation error. As a test problem for our numerical method, we consider a European cash-or-nothing call option. To show the effect of the adaptive stepping strategy, we calculate option price and its Greeks with various tolerances. Several numerical results confirm that the proposed method is fast, accurate, and practical in computing option price and the Greeks.

A practical finite difference method for the three-dimensional Black–Scholes equation

In this paper, we develop a fast and accurate numerical method for pricing of the three-asset equity-linked securities options. The option pricing model is based on the Black–Scholes partial differential equation. The model is discretized by using a non-uniform finite difference method and the resulting discrete equations are solved by using an operator splitting method. For fast and accurate calculation, we put more grid points near the singularity of the nonsmooth payoff function. To demonstrate the accuracy and efficiency of the proposed numerical method, we compare the results of the method with those from Monte Carlo simulation in terms of computational cost and accuracy. The numerical results show that the cost of the proposed method is comparable to that of the Monte Carlo simulation and it provides more stable hedging parameters such as the Greeks.

Chebyshev Wavelet Finite Difference Method: A New Approach for Solving Initial and Boundary Value Problems of Fractional Order

A new method based on a hybrid of Chebyshev wavelets and finite difference methods is introduced for solving linear and nonlinear fractional differential equations. The useful properties of the Chebyshev wavelets and finite difference method are utilized to reduce the computation of the problem to a set of linear or nonlinear algebraic equations. This method can be considered as a nonuniform finite difference method. Some examples are given to verify and illustrate the efficiency and simplicity of the proposed method.

A-SiC:H anti-resonant layer ARROW waveguides

Over the last decades, anti-resonant reflecting optical waveguides (ARROW) have been used in different integrated optics applications. In this type of waveguide, light confinement is partially achieved through an anti-resonant reflection. In this work, the simulation, fabrication and characterization of ARROW waveguides using dielectric films deposited by a plasma-enhanced chemical vapor deposition (PECVD) technique, at low temperatures ( ∼ 300 °C), are presented. Silicon oxynitride (SiO x N y ) films were used as core and second cladding layers and amorphous hydrogenated silicon carbide (a-SiC:H) films as first cladding layer. Furthermore, numerical simulations were performed using homemade routines based on two computational methods: the transfer matrix method (TMM) for the determination of the optimum thickness of the Fabry-Perot layers; and the non-uniform finite difference method (NU-FDM) for 2D design and determination of the maximum width that yields single-mode operation. The utilization of a silicon carbide anti-resonant layer resulted in low optical attenuations, which is due to the high refractive index difference between the core and this layer. Finally, for comparison purposes, optical waveguides using titanium oxide (TiO 2 ) as the first ARROW layer were also fabricated and characterized.

Numerical methods for semiconductor heterostructures with band nonparabolicity

This article presents numerical methods for computing bound state energies and associated wave functions of three-dimensional semiconductor heterostructures with special interest in the numerical treatment of the effect of band nonparabolicity. A nonuniform finite difference method is presented to approximate a model of a cylindrical-shaped semiconductor quantum dot embedded in another semiconductor matrix. A matrix reduction method is then proposed to dramatically reduce huge eigenvalue systems to relatively very small subsystems. Moreover, the nonparabolic band structure results in a cubic type of nonlinear eigenvalue problems for which a cubic Jacobi–Davidson method with an explicit nonequivalence deflation method are proposed to compute all the desired eigenpairs. Numerical results are given to illustrate the spectrum of energy levels and the corresponding wave functions in rather detail.

Numerical analysis of transverse mode in gain-guided vertical cavity surface emitting lasers

The stability of the fundamental transverse mode in a multiquantum-well vertical cavity surface emitting laser (VSCEL) has been studied by simulation. A full self-consistent model is presented for the VCSELs including the refractive index and the gain changes due to temperature rise and carrier variation in the gain medium. The nonuniform finite difference method is employed to solve the two-dimensional wave equation and simulate the behaviour of the VCSELs on desktop computers. The limitation to the maximum single mode power stems from the appearance of the higher-order transverse mode for the broad-area lasers, and the gain reduction caused by heating the small-area lasers. The self-focusing resulting from the spatial hole burning has a strong impact on the fundamental mode size. The calculations show that there are optimum values for both the quantum-well number and the conductive path diameter. Based on the chosen parameters a laser with three quantum wells and about 6 µm diameter conductive path yields the largest single mode output power.

A new approach to the analysis of the soliton: Nonuniform finite difference beam propagation method

A finite difference beam propagation method with nonuniform discretization is presented and validated for the analysis of solitary waves. We verify the accuracy of our nonuniform finite difference beam propagation method by demonstrating the evolution of the well-known N soliton in nonlinear dispersive medium with and without higher-order perturbed terms. Comparisons with the standard beam propagation method for calculation of both bright and dark solitons are also given and show that the nonuniform finite difference method is good for saving computing time, while maintaining the desired accuracy.

Two-dimensional analysis of optical waveguides with a nonuniform finite difference method

A nonuniform finite difference method is presented for an analysis of arbitrarily-shaped optical waveguides. By introducing preconditioning of discretised coefficient matrices by similarity transformations, it has become possible to improve the accuracy of the solution without paying any penalty in terms of computing time. From the comparison between the exact analytical method and the nonuniform finite difference method for an analysis of planar slab waveguides, it has been clarified that the mesh refinement near the material interfaces plays a dominant role in determining the accuracy of solutions. Furthermore, the nonuniform finite difference method is used to model buried-channel waveguides and is compared with the effective-index method. It has been indicated that differences between the non-uniform finite difference method and the effective-index method have become significant when the channel dimensions in both directions become less than the propagation wavelength.

Modeling of graded-index channel waveguides using nonuniform finite difference method

A finite-difference method (FDM) with nonuniform discretization for the analysis of channel waveguides is presented. Application of the boundary conditions for either the quasi-TE or quasi-TM mode is illustrated. Flexible discretization of the grid structures minimizes memory size, resulting in much smaller computing time without sacrificing the accuracy of the solution. This nonuniform discretization FDM technique is used to model the well-guided small-mode-size Ti:LiNbO/sub 3/ waveguides. The model treats both finite and infinite source diffusion cases. Quasi-TM mode profiles and the corresponding eigenvalues are rigorously evaluated and the theoretical results agree very well with the experimental results.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>