scholarly journals Analytical Particular Solutions of Multiquadrics Associated with Polyharmonic Operators

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Chia-Cheng Tsai
Keyword(s):  
Laurent Series ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Numerical Results ◽  
Poisson’S Equation ◽  
Poisson's Equation ◽  
Particular Solutions

We derive two- and three-dimensional analytical particular solutions of multiquadrics (MQ) associated with the polyharmonic operators, named as the polyharmonic multiquadrics (PMQs). The methods of undetermined coefficients are constructed by observing the first few orders of the PMQs which are obtained by the symbolic software,Mathematica. By expanding the PMQs into the Laurent series, the unknown coefficients of the PMQs can be determined. The homogeneous parts of the PMQs are suitably arranged so that the PMQs are hierarchically unique and infinitely differentiable.Mathematicacodes are provided for obtaining the PMQs of arbitrary orders. The derived PMQs are validated by numerical solutions for Poisson’s equation. Numerical results indicate that the solutions obtained by the PMQs are more accurate than those by the MQ.

scholarly journals A Family of Sixth-Order Compact Finite-Difference Schemes for the Three-Dimensional Poisson Equation

2010 ◽  
Vol 2010 ◽  
pp. 1-17 ◽  
Author(s):  
Yaw Kyei ◽  
John Paul Roop ◽  
Guoqing Tang
Keyword(s):  
Finite Difference ◽  
High Performance ◽  
Three Dimensional ◽  
Difference Schemes ◽  
Poisson’S Equation ◽  
Sixth Order ◽  
Finite Difference Schemes ◽  
Poisson's Equation ◽  
Compact Finite Difference ◽  
Compact Finite Difference Schemes

We derive a family of sixth-order compact finite-difference schemes for the three-dimensional Poisson's equation. As opposed to other research regarding higher-order compact difference schemes, our approach includes consideration of the discretization of the source function on a compact finite-difference stencil. The schemes derived approximate the solution to Poisson's equation on a compact stencil, and thus the schemes can be easily implemented and resulting linear systems are solved in a high-performance computing environment. The resulting discretization is a one-parameter family of finite-difference schemes which may be further optimized for accuracy and stability. Computational experiments are implemented which illustrate the theoretically demonstrated truncation errors.

A High-Accuracy Mechanical Quadrature Method for Solving the Axisymmetric Poisson's Equation

2017 ◽  
Vol 9 (2) ◽  
pp. 393-406 ◽  
Author(s):  
Hu Li ◽  
Jin Huang
Keyword(s):  
Three Dimensional ◽  
High Accuracy ◽  
Poisson’S Equation ◽  
Two Dimensional ◽  
Quadrature Method ◽  
Poisson's Equation ◽  
Mechanical Quadrature ◽  
Asymptotic Error ◽  
Accuracy Order ◽  
Mechanical Quadrature Method

AbstractIn this article, we consider the numerical solution for Poisson's equation in axisymmetric geometry. When the boundary condition and source term are axisymmetric, the problem reduces to solving Poisson's equation in cylindrical coordinates in the two-dimensional (r,z) region of the original three-dimensional domain S. Hence, the original boundary value problem is reduced to a two-dimensional one. To make use of the Mechanical quadrature method (MQM), it is necessary to calculate a particular solution, which can be subtracted off, so that MQM can be used to solve the resulting Laplace problem, which possesses high accuracy order and low computing complexities. Moreover, the multivariate asymptotic error expansion of MQM accompanied with for all mesh widths hi is got. Hence, once discrete equations with coarse meshes are solved in parallel, the higher accuracy order of numerical approximations can be at least by the splitting extrapolation algorithm (SEA). Meanwhile, a posteriori asymptotic error estimate is derived, which can be used to construct self-adaptive algorithms. The numerical examples support our theoretical analysis.

Fluid Flow and Lateral Fluid Dispersion in Bounded Granular Beds

2002 ◽  
Author(s):  
Azita Soleymani ◽  
Eveliina Takasuo ◽  
Piroz Zamankhan ◽  
William Polashenski
Keyword(s):  
Reynolds Number ◽  
Porous Media ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Numerical Study ◽  
Fluid Velocity ◽  
Three Dimensional ◽  
Numerical Results ◽  
Transition To Turbulence ◽  
Wide Range

Results are presented from a numerical study examining the flow of a viscous, incompressible fluid through random packing of nonoverlapping spheres at moderate Reynolds numbers (based on pore permeability and interstitial fluid velocity), spanning a wide range of flow conditions for porous media. By using a laminar model including inertial terms and assuming rough walls, numerical solutions of the Navier-Stokes equations in three-dimensional porous packed beds resulted in dimensionless pressure drops in excellent agreement with those reported in a previous study (Fand et al., 1987). This observation suggests that no transition to turbulence could occur in the range of Reynolds number studied. For flows in the Forchheimer regime, numerical results are presented of the lateral dispersivity of solute continuously injected into a three-dimensional bounded granular bed at moderate Peclet numbers. Lateral fluid dispersion coefficients are calculated by comparing the concentration profiles obtained from numerical and analytical methods. Comparing the present numerical results with data available in the literature, no evidence has been found to support the speculations by others for a transition from laminar to turbulent regimes in porous media at a critical Reynolds number.

scholarly journals Efficient and accurate solver of the three-dimensional screened and unscreened Poisson's equation with generic boundary conditions

2012 ◽  
Vol 137 (13) ◽  
pp. 134108 ◽  
Author(s):  
Alessandro Cerioni ◽  
Luigi Genovese ◽  
Alessandro Mirone ◽  
Vicente Armando Sole
Keyword(s):  
Boundary Conditions ◽  
Three Dimensional ◽  
Poisson’S Equation ◽  
Poisson's Equation ◽  
Generic Boundary

scholarly journals Analytical and Numerical Solutions for the Thermal Problem in a Friction Clutch System

Computation ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 122
Author(s):  
Laith A. Sabri ◽  
Katarzyna Topczewska ◽  
Muhsin Jaber Jweeg ◽  
Oday I. Abdullah ◽  
Azher M. Abed
Keyword(s):  
Numerical Solution ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Numerical Results ◽  
Three Dimensional Model ◽  
Thermal Problem ◽  
Friction Clutch ◽  
Dry Conditions ◽  
The Difference ◽  
Element Technique

The dry friction clutch is an important part in vehicles, which has more than one function, but the most important function is to connect and disconnect the engine (driving part) with driven parts. This work presents a developed numerical solution applying a finite element technique in order to obtain results with high precision. A new three-dimensional model of a single-disc clutch operating in dry conditions was built from scratch. As the new model represents the real friction clutch including all details, the complexity in the geometry of the clutch is considered one of the difficulties that the researchers faced using the numerical solution. The thermal behaviour of the friction clutch during the slip phase was studied. Meanwhile, in the second part of this work, the transient thermal equations were derived from scratch to find the analytical solution for the thermal problem of a clutch disc in order to verify the numerical results. It was found, after comparison of the numerical results with analytical results, that the results of the numerical model are very accurate and the difference between them does not exceed 1%.

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