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 Brownian Swarm Dynamics and Burgers’ Equation with Higher Order Dispersion

Symmetry ◽  
2020 ◽  
Vol 13 (1) ◽  
pp. 57
Author(s):  
Max-Olivier Hongler
Keyword(s):  
Burgers Equation ◽  
Higher Order ◽  
Underlying Structure ◽  
Swarm Dynamics ◽  
Special Cases ◽  
Kink Waves ◽  
Systematic Derivation ◽  
Fifth Order ◽  
Higher Order Dispersion ◽  
Order Dispersion

The concept of ranked order probability distribution unveils natural probabilistic interpretations for the kink waves (and hence the solitons) solving higher order dispersive Burgers’ type PDEs. Thanks to this underlying structure, it is possible to propose a systematic derivation of exact solutions for PDEs with a quadratic nonlinearity of the Burgers’ type but with arbitrary dispersive orders. As illustrations, we revisit the dissipative Kotrweg de Vries, Kuramoto-Sivashinski, and Kawahara equations (involving third, fourth, and fifth order dispersion dynamics), which in this context appear to be nothing but the simplest special cases of this infinitely rich class of nonlinear evolutions.

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 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 Runge-Kutta Type Methods for Directly Solving Special Fourth-Order Ordinary Differential Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Kasim Hussain ◽  
Fudziah Ismail ◽  
Norazak Senu
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fourth Order ◽  
Order Conditions ◽  
Numerical Results ◽  
Type Method ◽  
First Order ◽  
New Methods ◽  
Runge Kutta ◽  
Fifth Order

A Runge-Kutta type method for directly solving special fourth-order ordinary differential equations (ODEs) which is denoted by RKFD method is constructed. The order conditions of RKFD method up to order five are derived; based on the order conditions, three-stage fourth- and fifth-order Runge-Kutta type methods are constructed. Zero-stability of the RKFD method is proven. Numerical results obtained are compared with the existing Runge-Kutta methods in the scientific literature after reducing the problems into a system of first-order ODEs and solving them. Numerical results are presented to illustrate the robustness and competency of the new methods in terms of accuracy and number of function evaluations.

Conservative Transport Schemes for Spherical Geodesic Grids: High-Order Flux Operators for ODE-Based Time Integration

2011 ◽  
Vol 139 (9) ◽  
pp. 2962-2975 ◽  
Author(s):  
William C. Skamarock ◽  
Almut Gassmann
Keyword(s):  
Fourth Order ◽  
Time Integration ◽  
Higher Order ◽  
Second Order ◽  
Time Step ◽  
Diffusion Term ◽  
The Third ◽  
Integration Schemes ◽  
Order Scheme ◽  
Runge Kutta

Higher-order finite-volume flux operators for transport algorithms used within Runge–Kutta time integration schemes on irregular Voronoi (hexagonal) meshes are proposed and tested. These operators are generalizations of third- and fourth-order operators currently used in atmospheric models employing regular, orthogonal rectangular meshes. Two-dimensional least squares fit polynomials are used to evaluate the higher-order spatial derivatives needed to cancel the leading-order truncation error terms of the standard second-order centered formulation. Positive definite or monotonic behavior is achieved by applying an appropriate limiter during the final Runge–Kutta stage within a given time step. The third- and fourth-order formulations are evaluated using standard transport tests on the sphere. The new schemes are more accurate and significantly more efficient than the standard second-order scheme and other schemes in the literature examined by the authors. The third-order formulation is equivalent to the fourth-order formulation plus an additional diffusion term that is proportional to the Courant number. An optimal value for the coefficient scaling this diffusion term is chosen based on qualitative evaluation of the test results. Improvements using the higher-order scheme in place of the traditional second-order centered approach are illustrated within 3D unstable baroclinic wave simulations produced using two global nonhydrostatic models employing spherical Voronoi meshes.

scholarly journals New Phase-Fitted and Amplification-Fitted Fourth-Order and Fifth-Order Runge-Kutta-Nyström Methods for Oscillatory Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
K. W. Moo ◽  
N. Senu ◽  
F. Ismail ◽  
M. Suleiman
Keyword(s):  
Fourth Order ◽  
Variable Coefficients ◽  
Phase Lag ◽  
Oscillatory Solutions ◽  
Numerical Tests ◽  
New Methods ◽  
Nyström Methods ◽  
Runge Kutta ◽  
Fifth Order ◽  
New Phase

Two new Runge-Kutta-Nyström (RKN) methods are constructed for solving second-order differential equations with oscillatory solutions. These two new methods are constructed based on two existing RKN methods. Firstly, a three-stage fourth-order Garcia’s RKN method. Another method is Hairer’s RKN method of four-stage fifth-order. Both new derived methods have two variable coefficients with phase-lag of order infinity and zero amplification error (zero dissipative). Numerical tests are performed and the results show that the new methods are more accurate than the other methods in the literature.

scholarly journals A New Optimized Runge-Kutta Pair for the Numerical Solution of the Radial Schrödinger Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-15
Author(s):  
Yonglei Fang ◽  
Qinghong Li ◽  
Qinghe Ming ◽  
Kaimin Wang
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Phase Lag ◽  
Order Method ◽  
First Derivative ◽  
New Methods ◽  
Radial Schrödinger Equation ◽  
Runge Kutta ◽  
Fourth Order Method ◽  
Fifth Order

A new embedded pair of explicit modified Runge-Kutta (RK) methods for the numerical integration of the radial Schrödinger equation is presented. The two RK methods in the pair have algebraic orders five and four, respectively. The two methods of the embedded pair are derived by nullifying the phase lag, the first derivative of the phase lag of the fifth-order method, and the phase lag of the fourth-order method. Nu merical experiments show the efficiency and robustness of our new methods compared with some well-known integrators in the literature.

scholarly journals Study of Numerical Solution of Fourth Order Ordinary Differential Equations by fifth order Runge-Kutta Method

2019 ◽  
pp. 230-238
Author(s):  
Najmuddin Ahamad ◽  
Shiv Charan
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fourth Order ◽  
Initial Value Problems ◽  
Computational Cost ◽  
Kutta Method ◽  
Initial Value ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
Fifth Order

In this paper we present fifth order Runge-Kutta method (RK5) for solving initial value problems of fourth order ordinary differential equations. In this study RK5 method is quite efficient and practically well suited for solving boundary value problems. All mathematical calculation performed by MATLAB software for better accuracy and result. The result obtained, from numerical examples, shows that this method more efficient and accurate. These methods are preferable to some existing methods because of their simplicity, accuracy and less computational cost involved.

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