Comparing numerical methods for Boussinesq equation model problem

There are some classes of methods to solve the initial-value problem for the ODEs of the second order. Recently among of them are developed the numerical methods, which are using in the application of computer technology. By taking into account the wide application of the numerical methods, here has investigated the numerical solution of the above-mentioned problem. For this aim here has constructed the multistep hybrid method with the special structure, which has been applied to solve the initial-value problem of the ODEs of the second order. Given some recommendation to choosing of the suitable methods for solving above named problem and also, have found some bounders imposed on the coefficients of the convergence methods. Constructed specific methods solve the initial-value problem for ODEs of the second order. The received theoretical results have been illustrated by using some concrete methods, which have applied to solve model problem for ODEs of the second order

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ROBUST NUMERICAL METHODS FOR BOUNDARY‐LAYER EQUATIONS FOR A MODEL PROBLEM OF FLOW OVER A SYMMETRIC CURVED SURFACE

Mathematical Modelling and Analysis ◽

10.3846/13926292.2006.9637324 ◽

2006 ◽

Vol 11 (4) ◽

pp. 365-378

Author(s):

A. R. Ansari ◽

B. Hossain ◽

B. Koren ◽

G. I. Shishkin

Keyword(s):

Boundary Layer ◽

Numerical Methods ◽

Numerical Method ◽

Original Problem ◽

Model Problem ◽

Curved Surface ◽

Finite Domain ◽

Third Order ◽

Boundary Layer Equations ◽

Self Similar

We investigate the model problem of flow of a viscous incompressible fluid past a symmetric curved surface when the flow is parallel to its axis. This problem is known to exhibit boundary layers. Also the problem does not have solutions in closed form, it is modelled by boundary‐layer equations. Using a self‐similar approach based on a Blasius series expansion (up to three terms), the boundary‐layer equations can be reduced to a Blasius‐type problem consisting of a system of eight third‐order ordinary differential equations on a semi‐infinite interval. Numerical methods need to be employed to attain the solutions of these equations and their derivatives, which are required for the computation of the velocity components, on a finite domain with accuracy independent of the viscosity v, which can take arbitrary values from the interval (0,1]. To construct a robust numerical method we reduce the original problem on a semi‐infinite axis to a problem on the finite interval [0, K], where K = K(N) = ln N. Employing numerical experiments we justify that the constructed numerical method is parameter robust.

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A Comparison Study of Numerical Methods for Compressible Two-Phase Flows

Advances in Applied Mathematics and Mechanics ◽

10.4208/aamm.oa-2016-0084 ◽

2017 ◽

Vol 9 (5) ◽

pp. 1111-1132 ◽

Cited By ~ 2

Author(s):

Jianyu Lin ◽

Hang Ding ◽

Xiyun Lu ◽

Peng Wang

Keyword(s):

Numerical Methods ◽

Numerical Experiments ◽

Mass Conservation ◽

Detailed Comparison ◽

Equation Model ◽

Bubble Collapse ◽

Comparison Study ◽

Two Phase Flows ◽

Two Phase ◽

Interface Position

AbstractIn this article a comparison study of the numerical methods for compressible two-phase flows is presented. Although many numerical methods have been developed in recent years to deal with the jump conditions at the fluid-fluid interfaces in compressible multiphase flows, there is a lack of a detailed comparison of these methods. With this regard, the transport five equation model, the modified ghost fluid method and the cut-cell method are investigated here as the typical methods in this field. A variety of numerical experiments are conducted to examine their performance in simulating inviscid compressible two-phase flows. Numerical experiments include Richtmyer-Meshkov instability, interaction between a shock and a rectangle SF6 bubble, Rayleigh collapse of a cylindrical gas bubble in water and shock-induced bubble collapse, involving fluids with small or large density difference. Based on the numerical results, the performance of the method is assessed by the convergence order of the method with respect to interface position, mass conservation, interface resolution and computational efficiency.

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