High-order analytical solutions of the equations of relative motion with elliptic reference orbit

2014 ◽  
Vol 44 (6) ◽  
pp. 646-655 ◽  
Author(s):  
HanLun LEI ◽  
Bo XU
Keyword(s):  
Relative Motion ◽  
Analytical Solutions ◽  
High Order ◽  
Reference Orbit

scholarly journals Piecewise Approximate Analytical Solutions of High-Order Singular Perturbation Problems with a Discontinuous Source Term

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
Essam R. El-Zahar
Keyword(s):  
Asymptotic Expansion ◽  
Singular Perturbation ◽  
Analytical Solutions ◽  
Source Term ◽  
High Order ◽  
Singular Perturbation Problems ◽  
Weakly Coupled System ◽  
Approximate Analytical Solutions ◽  
Discontinuous Source Term ◽  
Perturbation Problems

A reliable algorithm is presented to develop piecewise approximate analytical solutions of third- and fourth-order convection diffusion singular perturbation problems with a discontinuous source term. The algorithm is based on an asymptotic expansion approximation and Differential Transform Method (DTM). First, the original problem is transformed into a weakly coupled system of ODEs and a zero-order asymptotic expansion of the solution is constructed. Then a piecewise smooth solution of the terminal value reduced system is obtained by using DTM and imposing the continuity and smoothness conditions. The error estimate of the method is presented. The results show that the method is a reliable and convenient asymptotic semianalytical numerical method for treating high-order singular perturbation problems with a discontinuous source term.

Computation of analytical solutions of the relative motion about a Keplerian elliptic orbit

2012 ◽  
Vol 81 (1) ◽  
pp. 186-199 ◽  
Author(s):  
Yuan Ren ◽  
Josep J. Masdemont ◽  
Manuel Marcote ◽  
Gerard Gómez
Keyword(s):  
Relative Motion ◽  
Analytical Solutions ◽  
Elliptic Orbit

A High Order Time Advancement Scheme for Prediction of Solidification Processes

2010 ◽  
Vol 297-301 ◽  
pp. 779-784 ◽  
Author(s):  
A. Abbasnejad ◽  
M.J. Maghrebi ◽  
H. Basirat Tabrizi
Keyword(s):  
Analytical Solutions ◽  
Second Order ◽  
High Order ◽  
Kutta Method ◽  
Third Order ◽  
Runge Kutta Method ◽  
Solidification Processes ◽  
Order Scheme ◽  
High Order Time ◽  
Good Agreement

The aim of this study is the simulation of alloys and pure materials solidification. A third order compact Runge-Kutta method and second order scheme are used for time advancement and space derivative modeling. The results are compared with analytical and semi-analytical solutions and show very good agreement.

scholarly journals CWENO Finite-Volume Interface Capturing Schemes for Multicomponent Flows Using Unstructured Meshes

2021 ◽  
Vol 89 (3) ◽  
Author(s):  
Panagiotis Tsoutsanis ◽  
Ebenezer Mayowa Adebayo ◽  
Adrian Carriba Merino ◽  
Agustin Perez Arjona ◽  
Martin Skote
Keyword(s):  
Finite Volume ◽  
Analytical Solutions ◽  
Unstructured Meshes ◽  
Accurate Solution ◽  
High Order ◽  
Experimental Results ◽  
Test Problems ◽  
Interface Capturing ◽  
Multicomponent Flows ◽  
Stringent Test

AbstractIn this paper we extend the application of unstructured high-order finite-volume central-weighted essentially non-oscillatory (CWENO) schemes to multicomponent flows using the interface capturing paradigm. The developed method achieves high-order accurate solution in smooth regions, while providing oscillation free solutions at discontinuous regions. The schemes are inherently compact in the sense that the central stencils employed are as compact as possible, and that the directional stencils are reduced in size, therefore simplifying their implementation. Several parameters that influence the performance of the schemes are investigated, such as reconstruction variables and their reconstruction order. The performance of the schemes is assessed under a series of stringent test problems consisting of various combinations of gases and liquids, and compared against analytical solutions, computational and experimental results available in the literature. The results obtained demonstrate the robustness of the new schemes for several applications, as well as their limitations within the present interface-capturing implementation.

A Generalized Sub-Equation Expansion Method and Some Analytical Solutions to the Inhomogeneous Higher-Order Nonlinear Schrödinger Equation

2008 ◽  
Vol 63 (12) ◽  
pp. 763-777 ◽  
Author(s):  
Biao Li ◽  
Yong Chen ◽  
Yu-Qi Li
Keyword(s):  
Solitary Wave ◽  
Analytical Solutions ◽  
Elliptic Function ◽  
Schrodinger Equation ◽  
Expansion Method ◽  
High Order ◽  
Solitary Wave Solutions ◽  
Exact Analytical Solutions ◽  
Nonlinear Schrödinger ◽  
Wave Solutions

On the basis of symbolic computation a generalized sub-equation expansion method is presented for constructing some exact analytical solutions of nonlinear partial differential equations. To illustrate the validity of the method, we investigate the exact analytical solutions of the inhomogeneous high-order nonlinear Schrödinger equation (IHNLSE) including not only the group velocity dispersion, self-phase-modulation, but also various high-order effects, such as the third-order dispersion, self-steepening and self-frequency shift. As a result, a broad class of exact analytical solutions of the IHNLSE are obtained. From our results, many previous solutions of some nonlinear Schrödinger-type equations can be recovered by means of suitable selections of the arbitrary functions and arbitrary constants. With the aid of computer simulation, the abundant structure of bright and dark solitary wave solutions, combined-type solitary wave solutions, dispersion-managed solitary wave solutions, Jacobi elliptic function solutions and Weierstrass elliptic function solutions are shown by some figures.

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