scholarly journals Solitary Wave Formation from a Generalized Rosenau Equation

2016 ◽  
Vol 2016 ◽  
pp. 1-17 ◽  
Author(s):  
J. I. Ramos ◽  
C. M. García-López
Keyword(s):  
Wave Front ◽  
Fourth Order ◽  
Accurate Method ◽  
Amplitude Curve ◽  
Convective Term ◽  
Compact Finite Differences ◽  
Rosenau Equation ◽  
Spatial Derivatives ◽  
Fifth Order ◽  
The Waves

A generalized viscous Rosenau equation containing linear and nonlinear advective terms and mixed third- and fifth-order derivatives is studied numerically by means of an implicit second-order accurate method in time that treats the first-, second-, and fourth-order spatial derivatives as unknown and discretizes them by means of three-point, fourth-order accurate, compact finite differences. It is shown that the effect of the viscosity is to decrease the amplitude, curve the wave trajectory, and increase the number and width of the waves that emerge from an initial Gaussian condition, whereas the linear convective term pushes the wave front towards the downstream boundary. It is also shown that the effect of the nonlinear convective term is to increase the steepness of the leading wave front and the number of sawtooth waves that are generated behind it, while that of the first dispersive term is to increase the number of waves that break up from the initial condition as the coefficient that characterizes this term is decreased. It is also shown that, for reasons of stability, the second dispersion coefficient must be much smaller than the first one and its effects on wave propagation are relatively small.

scholarly journals Hexagonal Grid Computation of the Derivatives of the Solution to the Heat Equation by Using Fourth-Order Accurate Two-Stage Implicit Methods

2021 ◽  
Vol 5 (4) ◽  
pp. 203
Author(s):  
Suzan Cival Buranay ◽  
Nouman Arshad ◽  
Ahmed Hersi Matan
Keyword(s):  
Heat Equation ◽  
Fourth Order ◽  
Boundary Value ◽  
Time Variable ◽  
Implicit Methods ◽  
First Order ◽  
Mixed Derivatives ◽  
Hexagonal Grids ◽  
Spatial Derivatives ◽  
Derivatives Of

We give fourth-order accurate implicit methods for the computation of the first-order spatial derivatives and second-order mixed derivatives involving the time derivative of the solution of first type boundary value problem of two dimensional heat equation. The methods are constructed based on two stages: At the first stage of the methods, the solution and its derivative with respect to time variable are approximated by using the implicit scheme in Buranay and Arshad in 2020. Therefore, Oh4+τ of convergence on constructed hexagonal grids is obtained that the step sizes in the space variables x1, x2 and in time variable are indicated by h, 32h and τ, respectively. Special difference boundary value problems on hexagonal grids are constructed at the second stages to approximate the first order spatial derivatives and the second order mixed derivatives of the solution. Further, Oh4+τ order of uniform convergence of these schemes are shown for r=ωτh2≥116,ω>0. Additionally, the methods are applied on two sample problems.

scholarly journals PHYSICAL REQUIREMENTS FOR A TAKEOFF IN SURFING

2017 ◽  
pp. 30
Author(s):  
Akihiko Kimura ◽  
Taro Kakinuma
Keyword(s):  
Wave Front ◽  
Time Variation ◽  
Wave Breaking ◽  
Horizontal Component ◽  
Breaking Point ◽  
Front Face ◽  
The Waves

The conditions required for a takeoff in surfing, are discussed, with the waves simulated numerically, considering two types of wave breaking, i.e., a plunging type, and a spilling type. First, a surfer is required to obtain a sufficient value for the horizontal component of paddling speed, not to be overtaken by a wave peak. Second, when the surfer stops paddling, he needs to be floating at a location where the force on him is downward, along the wave front face. On the basis of both conditions, the time variation of the required value for the horizontal component of paddling speed, is evaluated for both the plunging-type, and spilling-type, cases. When the paddling speed is sufficient, the surfable area is larger in the former case, than in the latter, on the offshore side of the wave-breaking point.

scholarly journals The Arab Formation in central Abu Dhabi: 3-D reservoir architecture and static and dynamic modeling

GeoArabia ◽  
2003 ◽  
Vol 8 (1) ◽  
pp. 47-86 ◽  
Author(s):  
Jürgen Grötsch ◽  
Omar Suwaina ◽  
Ghiath Ajlani ◽  
Ahmed Taher ◽  
Reyad El-Khassawneh ◽  
...  
Keyword(s):  
Dynamic Modeling ◽  
Fourth Order ◽  
High Energy ◽  
Development Planning ◽  
Abu Dhabi ◽  
Small Scale ◽  
Reservoir Properties ◽  
Depositional Facies ◽  
Well Completion ◽  
Fifth Order

ABSTRACT A 3-D geological model of the Kimmeridgian-Tithonian Manifa, Hith, Arab, and Upper Diyab formations in the area of the onshore Central Abu Dhabi Ridge was based on a newly established sequence stratigraphic, sedimentologic, and diagenetic model. It was part of an inter-disciplinary study of the large sour-gas reserves in Abu Dhabi that are mainly hosted by the Arab Formation. The model was used for dynamic evaluations and recommendations for further appraisal and development planning in the studied field. Fourth-order aggradational and progradational cycles are composed of small-scale fifth-order shallowing-upward cycles, mostly capped by anhydrite within the Arab-ABC. The study area is characterized by a shoreline progradation of the Arab Formation toward the east-northeast marked by high-energy oolitic/bioclastic grainstones of the Upper Arab-D and the Asab Oolite. The Arab-ABC, Hith, and Manifa pinch out toward the northeast. The strongly bioturbated Lower Arab-D is an intrashelf basinal carbonate ramp deposit, largely time-equivalent to the Arab-ABC. The deposition of the Manifa Formation over the Arab Formation was a major back-stepping event of the shallow-water platform before the onset of renewed progradation in the Early Cretaceous. Well productivity in the Arab-ABC is controlled mainly by thin, permeable dolomitic streaks in the fifth-order cycles at the base of the fourth-order cycles. This has major implications for reservoir management, well completion and stimulation, and development planning. Good reservoir properties have been preserved in the early diagenetic dolomitic streaks. In contrast, the reservoir properties of the Upper Arab-D oolitic/bioclastic grainstones deteriorate with depth due to burial diagenesis. A rock-type scheme was established because complex diagenetic overprinting prevented the depositional facies from being directly related to petrophysical properties. Special core analysis and the attribution of saturation functions to static and dynamic models were made on a cell-by-cell basis using the scheme and honoring the 3-D depositional facies and property model. The results demonstrated the importance of integrating sedimentological analysis and diagenesis with rock typing and static and dynamic modeling so as to enhance the predictive capabilities of subsurface models.

scholarly journals Linearization of symmetrical and asymmetrical two-way Doherty amplifier

2012 ◽  
Vol 25 (2) ◽  
pp. 161-170 ◽  
Author(s):  
Aleksandar Atanaskovic ◽  
Natasa Males-Ilic ◽  
Bratislav Milovanovic
Keyword(s):  
Fourth Order ◽  
Linearization Technique ◽  
The Third ◽  
Simulation Process ◽  
Input And Output ◽  
Carrier Cell ◽  
Fifth Order ◽  
Doherty Amplifier

The linearization effects on two-way Doherty amplifiers are presented in this paper. Symmetrical Doherty amplifier with the additional circuit for linearization has been realized and measurements of the linearization influence on the third- and fifth-order intermodulation products have been carried out. Asymmetrical Doherty amplifier has been designed and effects of the applied linearization technique have been considered through the simulation process. The linearization approach uses the fundamental signals? second harmonics and fourth-order nonlinear signals that are extracted at the output of the peaking cell, adjusted in amplitude and phase and injected at the input and output of the carrier cell in Doherty amplifier.

scholarly journals ANALYSIS OF DISPERSION ERRORS IN ACOUSTIC WAVE SIMULATIONS

2006 ◽  
Vol 5 (1) ◽  
pp. 62
Author(s):  
R. A. C. Germanos ◽  
L. F. De Souza
Keyword(s):  
Acoustic Wave ◽  
Fourth Order ◽  
Finite Differences ◽  
Acoustic Waves ◽  
Time Integration ◽  
Compact Scheme ◽  
High Order Accuracy ◽  
Discretization Schemes ◽  
Analysis Of Dispersion ◽  
The Waves

The governing equations of the acoustic problem are the compressible Euler equations. The discretization of these equations has to ensure that the acoustic waves are transported with non-dispersive and non-dissipative characteristics. In the present study numerical simulations of a standing acoustic wave are performed. Four different space discretization schemes are tested, namely, a second order finite-differences, a fourth order finitedifferences, a fourth order finite-differences compact scheme and a sixth order finite-differences compact scheme. The time integration is done with a fourth order Runge-Kutta scheme. The results obtained are compared with linearized analytical solutions. The influence of the dispersion on the simulation of a standing wave is analyzed. The results confirm that high order accuracy schemes can be more efficient for simulation of acoustic waves, especially the waves with high frequency.

A highly stable explicit scheme for a fourth-order nonlinear diffusion filter

2020 ◽  
Vol 11 (04) ◽  
pp. 2050030
Author(s):  
Mahipal Jetta
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Fourth Order ◽  
Nonlinear Diffusion ◽  
Splitting Scheme ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Diffusion Filter ◽  
Spatial Derivatives

The standard finite difference scheme (forward difference approximation for time derivative and central difference approximations for spatial derivatives) for fourth-order nonlinear diffusion filter allows very small time-step size to obtain stable results. The alternating directional implicit (ADI) splitting scheme such as Douglas method is highly stable but compromises accuracy for a relatively larger time-step size. In this paper, we develop [Formula: see text] stencils for the approximation of second-order spatial derivatives based on the finite pointset method. We then make use of these stencils for approximating the fourth-order partial differential equation. We show that the proposed scheme allows relatively bigger time-step size than the standard finite difference scheme, without compromising on the quality of the filtered image. Further, we demonstrate through numerical simulations that the proposed scheme is more efficient, in obtaining quality filtered image, than an ADI splitting scheme.

scholarly journals Diagonally Implicit Runge–Kutta Type Method for Directly Solving Special Fourth-Order Ordinary Differential Equations with Ill-Posed Problem of a Beam on Elastic Foundation

Algorithms ◽  
2018 ◽  
Vol 12 (1) ◽  
pp. 10 ◽  
Author(s):  
Nizam Ghawadri ◽  
Norazak Senu ◽  
Firas Adel Fawzi ◽  
Fudziah Ismail ◽  
Zarina Ibrahim
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Elastic Foundation ◽  
Fourth Order ◽  
Sixth Order ◽  
Test Problems ◽  
Type Method ◽  
Ill Posed ◽  
Runge Kutta ◽  
Fifth Order

In this study, fifth-order and sixth-order diagonally implicit Runge–Kutta type (DIRKT) techniques for solving fourth-order ordinary differential equations (ODEs) are derived which are denoted as DIRKT5 and DIRKT6, respectively. The first method has three and the another one has four identical nonzero diagonal elements. A set of test problems are applied to validate the methods and numerical results showed that the proposed methods are more efficient in terms of accuracy and number of function evaluations compared to the existing implicit Runge–Kutta (RK) methods.

scholarly journals Semi Implicit Hybrid Methods with Higher Order Dispersion for Solving Oscillatory Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
S. Z. Ahmad ◽  
F. Ismail ◽  
N. Senu ◽  
M. Suleiman
Keyword(s):  
Fourth Order ◽  
Absolute Stability ◽  
Hybrid Methods ◽  
Order Method ◽  
The Third ◽  
Dispersion Order ◽  
Runge Kutta ◽  
Fifth Order ◽  
Higher Order Dispersion ◽  
Order Dispersion

We constructed three two-step semi-implicit hybrid methods (SIHMs) for solving oscillatory second order ordinary differential equations (ODEs). The first two methods are three-stage fourth-order and three-stage fifth-order with dispersion order six and zero dissipation. The third is a four-stage fifth-order method with dispersion order eight and dissipation order five. Numerical results show that SIHMs are more accurate as compared to the existing hybrid methods, Runge-Kutta Nyström (RKN) and Runge-Kutta (RK) methods of the same order and Diagonally Implicit Runge-Kutta Nyström (DIRKN) method of the same stage. The intervals of absolute stability or periodicity of SIHM for ODE are also presented.

scholarly journals Fronts, domain walls and pulses in a generalized Ginzburg-Landau equation

1995 ◽  
Vol 38 (1) ◽  
pp. 77-97 ◽  
Author(s):  
Jinqiao Duan ◽  
Philip Holmes
Keyword(s):  
Domain Walls ◽  
Landau Equation ◽  
Order Term ◽  
Traveling Wave Solutions ◽  
Ginzburg Landau ◽  
Existence Proofs ◽  
Ginzburg Landau Equation ◽  
Pulse Type ◽  
Spatial Derivatives ◽  
Fifth Order

We discuss the existence and non-existence of front, domain wall and pulse type traveling wave solutions of a Ginzburg-Landau equation with cubic terms containing spatial derivatives and a fifth order term, in both subcritical and supercritical cases. Our results appear to be the first rigorous existence and non-existence proofs for the full equation with all possible terms derived from second order perturbation theory present.

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