scholarly journals Semi-Analytical Solution of Two-Dimensional Viscous Flow through Expanding/Contracting Gaps with Permeable Walls

2021 ◽  
Vol 26 (2) ◽  
pp. 41
Author(s):  
Mohammad Mehdi Rashidi ◽  
Mikhail A. Sheremet ◽  
Maryam Sadri ◽  
Satyaranjan Mishra ◽  
Pradyumna Kumar Pattnaik ◽  
...  
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Viscous Flow ◽  
Numerical Method ◽  
Asymptotic Method ◽  
Difference Method ◽  
Series Form ◽  
Advantages And Disadvantages ◽  
Adomian Decomposition ◽  
In Series

In this research, the analytical methods of the differential transform method (DTM), homotopy asymptotic method (HAM), optimal homotopy asymptotic method (OHAM), Adomian decomposition method (ADM), variation iteration method (VIM) and reproducing kernel Hilbert space method (RKHSM), and the numerical method of the finite difference method (FDM) for (analytical-numerical) simulation of 2D viscous flow along expanding/contracting channels with permeable borders are carried out. The solutions for analytical method are obtained in series form (and the series are convergent), while for the numerical method the solution is obtained taking into account approximation techniques of second-order accuracy. The OHAM and HAM provide an appropriate method for controlling the convergence of the discretization series and adjusting convergence domains, despite having a problem for large sizes of obtained results in series form; for instance, the size of the series solution for the DTM is very small for the same order of accuracy. It is hard to judge which method is the best and all of them have their advantages and disadvantages. For instance, applying the DTM to BVPs is difficult; however, solving BVPs with the HAM, OHAM and VIM is simple and straightforward. The extracted solutions, in comparison with the computational solutions (shooting procedure combined with a Runge–Kutta fourth-order scheme, finite difference method), demonstrate remarkable accuracy. Finally, CPU time, average error and residual error for different cases are presented in tables and figures.

Simulation of Rapid Heating in Fusion Reactor First Walls Using the Green’s Function Approach

1984 ◽  
Vol 106 (3) ◽  
pp. 486-490 ◽  
Author(s):  
A. M. Hassanein ◽  
G. L. Kulcinski
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Green’S Function ◽  
Green's Function ◽  
Difference Method ◽  
Moving Boundary ◽  
Rapid Heating ◽  
High Energy ◽  
Advantages And Disadvantages ◽  
The Finite Difference Method

The solution of the heat conduction probem in moving boundary conditions is very important in predicting accurate thermal behavior of materials when very high energy deposition is expected. Such high fluxes are encountered on first wall materials and other components in fusion reactors. A numerical method has been developed to solve this problem by the use of the Green’s function. A comparison is made between this method and a finite difference one. The comparison in the finite difference method is made with and without the variation of the thermophysical properties with temperature. The agreement between Green’s function and the finite difference method is found to be very good. The advantages and disadvantages of using the Green’s function method and the importance of the variation of material thermal properties with temperature are discussed.

A numerical method for a converging cylindrical shock

1957 ◽  
Vol 2 (2) ◽  
pp. 185-200 ◽  
Author(s):  
R. B. Payne
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Numerical Method ◽  
Compressible Fluid ◽  
Difference Method ◽  
Equations Of Motion ◽  
Symmetric Flow ◽  
Cylindrically Symmetric ◽  
Artificial Diffusion ◽  
The Finite Difference Method

The finite difference method due to Lax (1954) is used to solve the equations of motion for a cylindrically symmetric flow of a compressible fluid. In particular, a converging cylindrical shock is found to increase in strength in agreement with the formula of Chisnell (1957). The artificial diffusion introduced by the method causes the pressure to remain finite at the axis, but a reflected diverging shock is obtained.

scholarly journals An Efficient Compact Finite Difference Method for the Solution of the Gross-Pitaevskii Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Rongpei Zhang ◽  
Jia Liu ◽  
Guozhong Zhao
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Numerical Method ◽  
Difference Method ◽  
Three Dimensional ◽  
Gradient Flow ◽  
Pitaevskii Equation ◽  
Compact Finite Difference ◽  
Compact Finite Difference Method ◽  
Representation Technique

We present an efficient, unconditionally stable, and accurate numerical method for the solution of the Gross-Pitaevskii equation. We begin with an introduction on the gradient flow with discrete normalization (GFDN) for computing stationary states of a nonconvex minimization problem. Then we present a new numerical method, CFDM-AIF method, which combines compact finite difference method (CFDM) in space and array-representation integration factor (AIF) method in time. The key features of our methods are as follows: (i) the fourth-order accuracy in space andrth (r≥2) accuracy in time which can be achieved and (ii) the significant reduction of storage and CPU cost because of array-representation technique for efficient handling of exponential matrices. The CFDM-AIF method is implemented to investigate the ground and first excited state solutions of the Gross-Pitaevskii equation in two-dimensional (2D) and three-dimensional (3D) Bose-Einstein condensates (BECs). Numerical results are presented to demonstrate the validity, accuracy, and efficiency of the CFDM-AIF method.

scholarly journals Comparison of modified ADM and classical finite difference method for some third-order and fifth-order KdV equations

2021 ◽  
Vol 54 (1) ◽  
pp. 377-409
Author(s):  
Appanah Rao Appadu ◽  
Abey Sherif Kelil
Keyword(s):  
Exact Solution ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Numerical Experiment ◽  
Difference Method ◽  
Numerical Experiments ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Kdv Equations ◽  
Adomian Decomposition

Abstract The KdV equation, which appears as an asymptotic model in physical systems ranging from water waves to plasma physics, has been studied. In this paper, we are concerned with dispersive nonlinear KdV equations by using two reliable methods: Shehu Adomian decomposition method (STADM) and the classical finite difference method for solving three numerical experiments. STADM is constructed by combining Shehu’s transform and Adomian decomposition method, and the nonlinear terms can be easily handled using Adomian’s polynomials. The Shehu transform is used to accelerate the convergence of the solution series in most cases and to overcome the deficiency that is mainly caused by unsatisfied conditions in other analytical techniques. We compare the approximate and numerical results with the exact solution for the two numerical experiments. The third numerical experiment does not have an exact solution and we compare profiles from the two methods vs the space domain at some values of time. This study provides us with information about which of the two methods are effective based on the numerical experiment chosen. Knowledge acquired will enable us to construct methods for other related partial differential equations such as stochastic Korteweg-de Vries (KdV), KdV-Burgers, and fractional KdV equations.

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