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.

scholarly journals Mechanical Quadrature Method and Splitting Extrapolation for Solving Dirichlet Boundary Integral Equation of Helmholtz Equation on Polygons

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Hu Li ◽  
Yanying Ma
Keyword(s):  
Integral Equation ◽  
Helmholtz Equation ◽  
Boundary Integral Equation ◽  
Dirichlet Boundary ◽  
Boundary Integral ◽  
Quadrature Method ◽  
Mechanical Quadrature ◽  
Splitting Extrapolation ◽  
Accuracy Order ◽  
Mechanical Quadrature Method

We study the numerical solution of Helmholtz equation with Dirichlet boundary condition. Based on the potential theory, the problem can be converted into a boundary integral equation. We propose the mechanical quadrature method (MQM) using specific quadrature rule to deal with weakly singular integrals. Denote byhmthe mesh width of a curved edgeΓm  (m=1,…,d)of polygons. Then, the multivariate asymptotic error expansion of MQM accompanied withO(hm3)for all mesh widthshmis obtained. Hence, once discrete equations with coarse meshes are solved in parallel, the higher accuracy order of numerical approximations can be at leastO(hmax⁡5)by splitting extrapolation algorithm (SEA). A numerical example is provided to support our theoretical analysis.

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.

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 Evaluations on The Use Of 3d Terrestrial Laser Scanning Technology in Architectural Conservation Projects

2021 ◽  
Author(s):  
Saadet Armağan Güleç Korumaz ◽  
◽  
Büşra Kubiloğlu ◽  
Keyword(s):  
Cultural Heritage ◽  
Point Cloud ◽  
Laser Scanning ◽  
Three Dimensional ◽  
Terrestrial Laser Scanning ◽  
Field Studies ◽  
High Accuracy ◽  
Two Dimensional ◽  
Conservation Projects ◽  
Architectural Documentation

3D Laser Scanning technologies have proven to be significant way to architectural documentation studies. Due to these facilities, the use of technology in architectural documentation have become widespread day by day. Thanks to these technologies it is possible to get high accuracy and intense data in a short time compared to conventional methods. Therefore, this technology has increased the content and quality of conservation practices. The technology is mainly aimed at obtaining a three-dimensional model or two-dimensional layouts from a dense and detailed point cloud. Terrestrial Laser Scanning (TLS) does not only support simple CAD-based conservation projects, but also allows obtaining high-resolution plane pictures, art tours, three-dimensional mesh models, and two-dimensional maps. Besides these possibilities, high accuracy data on the morphological properties of the documented object can be obtained as a result of the analyses including point cloud. On the other hand, the technology gives possibility data to be shared in different environments and filtered data can be used online. Thus, different disciplines are able to easily access information. These features of technology add a different dimension to the studies in the field of cultural heritage and contribute to the digitalization of the heritage. In the scope of this study, evaluations are made regarding the innovations and usage possibilities brought by TLS technology to architectural documentation field based on the cultural heritage samples. In addition, within the scope of the study, trials were made on field studies for parameters that will affect data quality, accuracy and speed. In addition, within the scope of the study, some tests were made on field studies for parameters affecting data quality, accuracy and speed. With the obtained results, evaluations have been made to increase the usage potential of the technology today.

scholarly journals A Mechanical Quadrature Method for Solving Delay Volterra Integral Equation with Weakly Singular Kernels

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-12
Author(s):  
Li Zhang ◽  
Jin Huang ◽  
Yubin Pan ◽  
Xiaoxia Wen
Keyword(s):  
Integral Equation ◽  
Volterra Integral Equation ◽  
Richardson Extrapolation ◽  
Quadrature Method ◽  
Mechanical Quadrature ◽  
Proportional Delays ◽  
Weakly Singular ◽  
Uniqueness Of The Solution ◽  
Singular Kernels ◽  
Mechanical Quadrature Method

In this work, a mechanical quadrature method based on modified trapezoid formula is used for solving weakly singular Volterra integral equation with proportional delays. An improved Gronwall inequality is testified and adopted to prove the existence and uniqueness of the solution of the original equation. Then, we study the convergence and the error estimation of the mechanical quadrature method. Moreover, Richardson extrapolation based on the asymptotic expansion of error not only possesses a high accuracy but also has the posterior error estimate which can be used to design self-adaptive algorithm. Numerical experiments demonstrate the efficiency and applicability of the proposed method.

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