scholarly journals Achieving Seventh-Order Amplitude Accuracy in Leapfrog Integrations

2013 ◽  
Vol 141 (9) ◽  
pp. 3037-3051 ◽  
Author(s):  
Paul D. Williams
Keyword(s):  
Nonlinear System ◽  
Third Order ◽  
First Order ◽  
Time Stepping ◽  
The Third ◽  
Numerical Integrations ◽  
Seventh Order ◽  
Ocean Models ◽  
Stability And Accuracy ◽  
Fifth Order

Abstract The leapfrog time-stepping scheme makes no amplitude errors when integrating linear oscillations. Unfortunately, the Robert–Asselin filter, which is used to damp the computational mode, introduces first-order amplitude errors. The Robert–Asselin–Williams (RAW) filter, which was recently proposed as an improvement, eliminates the first-order amplitude errors and yields third-order amplitude accuracy. However, it has not previously been shown how to further improve the accuracy by eliminating the third- and higher-order amplitude errors. Here, it is shown that leapfrogging over a suitably weighted blend of the filtered and unfiltered tendencies eliminates the third-order amplitude errors and yields fifth-order amplitude accuracy. It is further shown that the use of a more discriminating (1, −4, 6, −4, 1) filter instead of a (1, −2, 1) filter eliminates the fifth-order amplitude errors and yields seventh-order amplitude accuracy. Other related schemes are obtained by varying the values of the filter parameters, and it is found that several combinations offer an appealing compromise of stability and accuracy. The proposed new schemes are tested in numerical integrations of a simple nonlinear system. They appear to be attractive alternatives to the filtered leapfrog schemes currently used in many atmosphere and ocean models.

scholarly journals Rogue Wave Solutions and Generalized Darboux Transformation for an Inhomogeneous Fifth-Order Nonlinear Schrödinger Equation

2017 ◽  
Vol 2017 ◽  
pp. 1-13
Author(s):  
N. Song ◽  
W. Zhang ◽  
P. Wang ◽  
Y. K. Xue
Keyword(s):  
Darboux Transformation ◽  
Rogue Wave ◽  
Rogue Waves ◽  
Third Order ◽  
First Order ◽  
The Third ◽  
Wave Solutions ◽  
Generalized Darboux Transformation ◽  
Rogue Wave Solutions ◽  
Fifth Order

The rogue wave solutions are discussed for an inhomogeneous fifth-order nonlinear Schrödinger equation, which describes the dynamics of a site-dependent Heisenberg ferromagnetic spin chain. Using the Darboux matrix, the generalized Darboux transformation is constructed and a recursive formula is derived. Based on the transformation, the first-order to the third-order rogue wave solutions are obtained. Then, the nonlinear dynamics of the first-order to the third-order rogue waves are studied on the basis of some free parameters. Several new structures of the rogue waves are found using numerical simulation. The conclusions will be a supportive tool to study the rogue waves better.

Intertextual Precision Expectations

2019 ◽  
pp. 107-116
Author(s):  
Karin Kukkonen
Keyword(s):  
Third Order ◽  
Structural Patterns ◽  
First Order ◽  
The Third ◽  
Order Probability

In the chapters that follow, the third-order probability design is developed. The third-order probability design revolves around how expectations about second- and first-order predictions are developed through structural patterns yielded by genre (III.1), textual gaps and shadow stories (III.2), and intertextual references to unfamiliar texts (III.3). The final chapter of the section, then, traces the tension between flexibility and constraint in probability designs.

WENO-TVD schemes for hyperbolic conservation laws

Analysis ◽  
2007 ◽  
Vol 27 (1) ◽  
Author(s):  
Yousef Hashem Zahran
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
Building Blocks ◽  
Second Order ◽  
Third Order ◽  
Weno Schemes ◽  
Hyperbolic Conservation ◽  
Systematic Assessment ◽  
The Third ◽  
Seventh Order

The purpose of this paper is twofold. Firstly we carry out a modification of the finite volume WENO (weighted essentially non-oscillatory) scheme of Titarev and Toro [14] and [15].This modification is done by using two fluxes as building blocks in spatially fifth order WENO schemes instead of the second order TVD flux proposed by Titarev and Toro [14] and [15]. These fluxes are the second order TVD flux [19] and the third order TVD flux [20].Secondly, we propose to use these fluxes as a building block in spatially seventh order WENO schemes. The numerical solution is advanced in time by the third order TVD Runge–Kutta method. A way to extend these schemes to general systems of nonlinear hyperbolic conservation laws, in one and two dimension is presented. Systematic assessment of the proposed schemes shows substantial gains in accuracy and better resolution of discontinuities, particularly for problems involving long time evolution containing both smooth and non-smooth features.

scholarly journals Positive Solutions for Third-Order Boundary-Value Problems with the Integral Boundary Conditions and Dependence on the First-Order Derivatives

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Yanping Guo ◽  
Fei Yang
Keyword(s):  
Fixed Point ◽  
Green Function ◽  
Boundary Conditions ◽  
Boundary Value ◽  
Integral Boundary Conditions ◽  
Third Order ◽  
Integral Boundary ◽  
First Order ◽  
The Third ◽  
Nonnegative Continuous Function

By using a fixed point theorem in a cone and the nonlocal third-order BVP's Green function, the existence of at least one positive solution for the third-order boundary-value problem with the integral boundary conditionsx′′′(t)+f(t,x(t),x′(t))=0,t∈J,x(0)=0,x′′(0)=0, andx(1)=∫01g(t)x(t)dtis considered, wherefis a nonnegative continuous function,J=[0,1], andg∈L[0,1].The emphasis here is thatfdepends on the first-order derivatives.

scholarly journals MODIFIED WAVE EQUATION FOR SPINLESS PARTICLES AND ITS SOLUTIONS IN AN EXTERNAL MAGNETIC FIELD

2013 ◽  
Vol 28 (06) ◽  
pp. 1350014 ◽  
Author(s):  
S. I. KRUGLOV
Keyword(s):  
Magnetic Field ◽  
Wave Equation ◽  
External Magnetic Field ◽  
Quantum Mechanical ◽  
Third Order ◽  
Modified Dispersion Relation ◽  
First Order ◽  
The Third ◽  
Uniform External Magnetic Field ◽  
Schrödinger Form

The wave equation for spinless particles with the Lorentz violating term is considered. We formulate the third-order in derivatives wave equation leading to the modified dispersion relation. The first-order formalism is considered and the density matrix is obtained. The Schrödinger form of equations is presented and the quantum-mechanical Hamiltonian is found. Exact solutions of the wave equation are obtained for particles in the constant and uniform external magnetic field. The change of the synchrotron radiation radius due to quantum gravity corrections is calculated.

Perturbation treatment of the Hartree–Fock equations for the ground states of the boron-like ions

1969 ◽  
Vol 47 (7) ◽  
pp. 699-705 ◽  
Author(s):  
C. S. Sharma ◽  
R. G. Wilson
Keyword(s):  
Ground State ◽  
Atomic System ◽  
Ground States ◽  
Great Accuracy ◽  
Second Order ◽  
Expectation Values ◽  
Third Order ◽  
First Order ◽  
Hartree Fock ◽  
The Third

The first-order Hartree–Fock and unrestricted Hartree–Fock equations for the ground state of a five electron atomic system are solved exactly. The solutions are used to evaluate the corresponding second-order energies exactly and the third-order energies with great accuracy. The first-order terms in the expectation values of 1/r, r, r2, and δ(r) are also calculated.

A perturbation calculation of properties of the 2 pπ state of HeH 2+

1957 ◽  
Vol 240 (1221) ◽  
pp. 274-283 ◽  
Keyword(s):  
Dipole Moment ◽  
Total Energy ◽  
Second Order ◽  
Wave Functions ◽  
Quadrupole Moments ◽  
Perturbation Calculation ◽  
Third Order ◽  
The Third ◽  
Fifth Order ◽  
Kinetic And Potential Energies

A perturbation calculation, valid in the limit of large separations, of various properties of the 2 pπ state of HeH 2+ is carried out. The total energy and the kinetic and potential energies are calculated to the fifth order, the dipole moment to the third order and the quadrupole moments to the second order and the results compared with those obtained using exact and variationally determined two-centre wave functions. Some results are also given for the 2 pπ u and 3 dπ g states of H + 2 and the influence of nuclear symmetry at large separations is briefly discussed.

scholarly journals Solitons, Rogues and Interaction Behaviors of Third-Order Nonlinear Schrodinger Equation

2022 ◽  
Author(s):  
Ren Bo ◽  
Shi Kai-Zhong ◽  
Shou-Feng Shen ◽  
Wang Guo-Fang ◽  
Peng Jun-Da ◽  
...  
Keyword(s):  
Bilinear Form ◽  
Rogue Wave ◽  
Soliton Solutions ◽  
Ultrashort Pulses ◽  
Rogue Waves ◽  
Third Order ◽  
First Order ◽  
The Third ◽  
Femtosecond Regime ◽  
Wave Limit

Abstract In this paper, we investigate the third-order nonlinear Schr\"{o}dinger equation which is used to describe the propagation of ultrashort pulses in the subpicosecond or femtosecond regime. Based on the independent transformation, the bilinear form of the third-order NLSE is constructed. The multiple soliton solutions are constructed by solving the bilinear form. The multi-order rogue waves and interaction between one-soliton and first-order rogue wave are obtained by the long wave limit in multi-solitons. The dynamics of the first-order rogue wave, second-order rogue wave and interaction between one-soliton and first-order rogue wave are presented by selecting the appropriate parameters. In particular parameters, the positions and the maximum of amplitude of rogue wave can be confirmed by the detail calculations.PACS numbers: 02.30.Ik, 05.45.Yv.

scholarly journals Influence on The Level Riders Motorcycle Accident Traffic with Traffic Methods of Conflict Technique (TCT) (Case Study: Jalan Raya Narogong Bekasi City)

2020 ◽  
Vol 2 (2) ◽  
pp. 184-189
Author(s):  
Andri Irfan Rifai ◽  
Finsa Aziz Fernanda
Keyword(s):  
Traffic Accidents ◽  
Observation Data ◽  
Third Order ◽  
First Order ◽  
The Third ◽  
Motorcycle Accident ◽  
Transportation Sector ◽  
Side Collision ◽  
Traffic Rules

The increasing number of traffic accidents can be caused by drivers, vehicles, highways, and the environment. In Indonesia, traffic accidents become one of the problems in the transportation sector. Prevention is done during this time to anticipate accidents only based on the data of the accident quantity that has occurred. Though factors or incidents that can cause accidents to become the biggest contributor in the event of accidents. For example, driving a vehicle in an unorderly manner, the pace of the vehicle with the above-average velocity set traffic rules, and sudden vehicle maneuvers. This research is done by identifying and analyzing the behavior of motorcyclists who affect accidents and applying TCT methods to observation data at points that become potential locations Against accidents. The research location is on the Narogong Highway which is divided into 2 segments. In Segment 1 begins at junction four Cipendawa (after the flyover Simpang Cipendawa) until the junction of the three Gg. Sawo (Bantar Gebang Market). Next, in Segment 2 starts from junction three of Gg. Sawo (Bantar Gebang Market) until the three houses of Vida housing. The results showed that the research location had potential that could cause the accident to be front-side on the first order, collision front-front on the second-order, and side-by-side collision on the third order. The speed of vehicles has an impact on accidents.

scholarly journals CALCULATION OF THE HIGHER-ORDER AXIAL SPHERICAL ABERRATIONS OF A HIGH-APERTURE FOCUSING HOLOGRAPHIC OPTICAL ELEMENT WITH THE CORRECTED THIRD-ORDER SPHERICAL ABERRATION

2018 ◽  
Vol 42 (1) ◽  
pp. 44-53
Author(s):  
Yu. Ts. Batomunkuev ◽  
A. A. Dianova
Keyword(s):  
Spherical Aberration ◽  
Visible Spectrum ◽  
Optical Element ◽  
Higher Order ◽  
Point Sources ◽  
Third Order ◽  
The Third ◽  
Holographic Optical Element ◽  
Seventh Order ◽  
Spectral Ranges

Results of calculating the radius of higher-order spherical aberrations (fifth, seventh and ninth orders) of a high-aperture focusing holographic optical element (HOE) with corrected third-order spherical aberration in the operating spectral range are discussed. As examples, high-aperture axial HOEs with relative apertures close to 1:1 in specified spectral ranges are considered. Coordinates of the point sources of a divergent reference wave and a convergent object wave of the HOE are given. It is shown that when imaging a point source emitting in the 0.250-0.281-µm and 0.500- 0.563-µm spectral ranges, the use of an HOE in the first and second diffraction orders makes it is possible to correct the third-order spherical aberration on two wavelengths and the fifth- and seventh-order spherical aberrations on one wavelength. Note that these visible-spectrum wavelengths are different from the HOE's recording wavelength of 0.532 µm.

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