scholarly journals Acquiring the Symplectic Operator Based on Pure Mathematical Derivation then Verifying It in the Intrinsic Problem of Nanodevices

Symmetry ◽  
2019 ◽  
Vol 11 (11) ◽  
pp. 1383
Author(s):  
Nie ◽  
Gui ◽  
Chen
Keyword(s):  
Finite Difference Time Domain ◽  
Intrinsic Properties ◽  
Hamilton Equation ◽  
Dispersion Error ◽  
One Dimensional ◽  
Particle Wave Function ◽  
The One ◽  
Mathematical Derivation ◽  
Difference Time ◽  
Total Energy Distribution

The symplectic algorithm can maintain the symplectic structure and intrinsic properties of the system, its cumulative error is small and suitable for multi-step calculation. At present, the widely accepted symplectic operators are obtained by solving the Hamilton equation based on artificial definitions and assumptions in advance. There are inevitable dispersion errors. We solve the equation by pure mathematical derivation without any artificial limitations and assumptions. The way to accurately obtain high-precision symplectic operators greatly reduces the dispersion error from the beginning. The numerical solution of the one-dimensional Schrödinger equation for describing the intrinsic problem of nanodevices is used as an application environment to compare the total energy distribution of the particle wave function in the box, thus verifying the properties of the Symplectic Operator based on Pure Mathematical Derivation by comparing with Finite-Difference Time-Domain (FDTD) and the widely accepted symplectic operator.

scholarly journals Non-Standard and Numerov Finite Difference Schemes for Finite Difference Time Domain Method to Solve One- Dimensional Schrödinger Equation

2018 ◽  
Vol 2 (1) ◽  
pp. 27
Author(s):  
Lily Maysari Angraini ◽  
I Wayan Sudiarta
Keyword(s):  
Finite Difference ◽  
Harmonic Oscillator ◽  
Time Domain ◽  
Finite Difference Time Domain ◽  
Fdtd Method ◽  
Finite Difference Schemes ◽  
Potential Wells ◽  
One Dimensional ◽  
The One ◽  
Difference Time

<span>The purpose of  this paper is to show some improvements of the finite-difference time domain (FDTD) method using Numerov and non-standard finite difference (NSFD) schemes for solving the one-dimensional Schr</span><span>ö</span><span>dinger equation. Starting with results of the unmodified FDTD method, Numerov-FD and NSFD are applied iteratively to produce more accurate results for eigen energies and wavefunctios. Three potential wells, infinite square well, harmonic oscillator and Poschl-Teller, are used to compare results of FDTD calculations. Significant improvements in the results for the infinite square potential and the harmonic oscillator potential are found using Numerov-NSFD scheme, and for Poschl-Teller potential are found using Numerov scheme.</span>

scholarly journals One-Step Leapfrog LOD-BOR-FDTD Algorithm with CPML Implementation

2016 ◽  
Vol 2016 ◽  
pp. 1-8
Author(s):  
Yi-Gang Wang ◽  
Yun Yi ◽  
Bin Chen ◽  
Hai-Lin Chen ◽  
Kang Luo ◽  
...  
Keyword(s):  
Time Domain ◽  
Finite Difference Time Domain ◽  
Algebraic Manipulation ◽  
Body Of Revolution ◽  
One Dimensional ◽  
Electric And Magnetic Fields ◽  
One Step ◽  
Cartesian Coordinate ◽  
The One ◽  
Difference Time

An unconditionally stable one-step leapfrog locally one-dimensional finite-difference time-domain (LOD-FDTD) algorithm towards body of revolution (BOR) is presented. The equations of the proposed algorithm are obtained by the algebraic manipulation of those used in the conventional LOD-BOR-FDTD algorithm. The equations forz-direction electric and magnetic fields in the proposed algorithm should be treated specially. The new algorithm obtains a higher computational efficiency while preserving the properties of the conventional LOD-BOR-FDTD algorithm. Moreover, the convolutional perfectly matched layer (CPML) is introduced into the one-step leapfrog LOD-BOR-FDTD algorithm. The equation of the one-step leapfrog CPML is concise. Numerical results show that its reflection error is small. It can be concluded that the similar CPML scheme can also be easily applied to the one-step leapfrog LOD-FDTD algorithm in the Cartesian coordinate system.

scholarly journals Analytical solution of one-dimensional Pennes’ bioheat equation

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 1084-1092
Author(s):  
Hongyun Wang ◽  
Wesley A. Burgei ◽  
Hong Zhou
Keyword(s):  
Analytical Solution ◽  
Radiofrequency Energy ◽  
Bioheat Equation ◽  
Surface Cooling ◽  
One Dimensional ◽  
The Fourier Transform ◽  
The One ◽  
Parameter Values ◽  
Mathematical Derivation ◽  
Effect Of Surface

Abstract Pennes’ bioheat equation is the most widely used thermal model for studying heat transfer in biological systems exposed to radiofrequency energy. In their article, “Effect of Surface Cooling and Blood Flow on the Microwave Heating of Tissue,” Foster et al. published an analytical solution to the one-dimensional (1-D) problem, obtained using the Fourier transform. However, their article did not offer any details of the derivation. In this work, we revisit the 1-D problem and provide a comprehensive mathematical derivation of an analytical solution. Our result corrects an error in Foster’s solution which might be a typo in their article. Unlike Foster et al., we integrate the partial differential equation directly. The expression of solution has several apparent singularities for certain parameter values where the physical problem is not expected to be singular. We show that all these singularities are removable, and we derive alternative non-singular formulas. Finally, we extend our analysis to write out an analytical solution of the 1-D bioheat equation for the case of multiple electromagnetic heating pulses.

scholarly journals From Time-Collocated to Leapfrog Fundamental Schemes for ADI and CDI FDTD Methods

Axioms ◽  
2022 ◽  
Vol 11 (1) ◽  
pp. 23
Author(s):  
Eng Leong Tan
Keyword(s):  
Finite Difference ◽  
Time Domain ◽  
Finite Difference Time Domain ◽  
Fdtd Method ◽  
Unconditionally Stable ◽  
Alternating Direction Implicit ◽  
One Dimensional ◽  
Alternating Direction ◽  
Fdtd Methods ◽  
Difference Time

The leapfrog schemes have been developed for unconditionally stable alternating-direction implicit (ADI) finite-difference time-domain (FDTD) method, and recently the complying-divergence implicit (CDI) FDTD method. In this paper, the formulations from time-collocated to leapfrog fundamental schemes are presented for ADI and CDI FDTD methods. For the ADI FDTD method, the time-collocated fundamental schemes are implemented using implicit E-E and E-H update procedures, which comprise simple and concise right-hand sides (RHS) in their update equations. From the fundamental implicit E-H scheme, the leapfrog ADI FDTD method is formulated in conventional form, whose RHS are simplified into the leapfrog fundamental scheme with reduced operations and improved efficiency. For the CDI FDTD method, the time-collocated fundamental scheme is presented based on locally one-dimensional (LOD) FDTD method with complying divergence. The formulations from time-collocated to leapfrog schemes are provided, which result in the leapfrog fundamental scheme for CDI FDTD method. Based on their fundamental forms, further insights are given into the relations of leapfrog fundamental schemes for ADI and CDI FDTD methods. The time-collocated fundamental schemes require considerably fewer operations than all conventional ADI, LOD and leapfrog ADI FDTD methods, while the leapfrog fundamental schemes for ADI and CDI FDTD methods constitute the most efficient implicit FDTD schemes to date.

Effect of Metal Intercalating on Transmission Characteristics of One-Dimensional Photonic Crystal

2011 ◽  
Vol 418-420 ◽  
pp. 679-683
Author(s):  
Bei Jia He ◽  
Xin Yi Chen ◽  
Jian Bo Wang ◽  
Jun Lu ◽  
Jian Chang ◽  
...  
Keyword(s):  
Photonic Crystal ◽  
Plasma Frequency ◽  
Fdtd Method ◽  
Transmission Characteristics ◽  
Material Characteristic ◽  
One Dimensional ◽  
Dimensional Photonic Crystal ◽  
Simulation Results ◽  
The One ◽  
Difference Time

To expand the bandgap's width of the one-dimensional photonic crystal, a crystal named SiO2/Metal/MgF2 is formed by joining some metals into the crystal SiO2/MgF2. Furthermore the Finite Difference Time Domain (FDTD) method is used to explore the metals' influence on the crystal's transmission characteristics. The simulation results show that the metals joined could expand the width of the one-dimensional photonic crystal's bandgap effectively and the bandgap's width increases when the metals' thickness increases. Meanwhile the bandgap's characteristic is affected by the metals' material-characteristic. The higher the plasma frequency is, the wider the bandgap's width will be and the more the number of the bandgaps will be. On the other hand, the metals' damping frequency has no significant effect on the bandgap, but would make the bandgap-edge's transmittance decrease slightly.

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