Higher-order Cartesian Grid Based Finite Difference Methods for Elliptic Equations on Irregular Domains and Interface Problems and Their Applications PDF Download
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Author: Yaw Kyei
Publisher:
ISBN:
Category :
Languages : en
Pages : 151
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Book Description
Keywords: Irregular Domains, Cartesian Grid, boundary control, optimization, Elliptic Equations, open and closed loop eigenvalues, Semi-discretization, state feedback control, Higher-Order finite difference method, standard compact 9-point stencil, Heat Equation, Elliptic Interface Problems, continuation of solution.
Author: Yaw Kyei
Publisher:
ISBN:
Category :
Languages : en
Pages : 151
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Book Description
Keywords: Irregular Domains, Cartesian Grid, boundary control, optimization, Elliptic Equations, open and closed loop eigenvalues, Semi-discretization, state feedback control, Higher-Order finite difference method, standard compact 9-point stencil, Heat Equation, Elliptic Interface Problems, continuation of solution.
Author:
Publisher:
ISBN:
Category :
Languages : en
Pages :
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Book Description
This thesis describes higher-order finite difference methods for solving elliptic equations on irregular domains with general boundary conditions and the corresponding elliptic interface problems. We develop second and fourth order methods for two and three dimensions using uniform Cartesian grids. However, with an irregular domain we cannot apply the standard finite difference schemes directly at the grid points near the boundary and therefore some treatment is required in order to use the uniform Cartesian grids. Our approach involves modifying the standard finite difference schemes. In particular, we use the standard five-point and standard compact nine-point stencil schemes for the second and fourth order methods, respectively. That is, on the standard stencils that contain the boundary, we carry out the modification by applying the continuation of solution from the inside of domain to the outside. The method of continuation of solution uses Taylor series expansion of the solution about selected boundary points, the equation and the boundary values of the local and their nearby boundary points. Naturally, second and fourth order Taylor series expansions about the boundary points are used for the second and fourth order methods respectively. Our methods have a unified and an effective approach to deal with general boundary conditions and capture the boundary and its local geometrical properties by the level set function and the local coordinate system at the boundary points. The resulting finite difference system matrices of our methods remain symmetric positive definite and maintain the sparsity of the standard finite difference schemes. As part of our main objective, we apply our fourth order method to semi-discretize the corresponding parabolic equation in space on the irregular domain and obtain an ODE system. The validity and effectiveness of the proposed method is clearly demonstrated through the computation of eigenvalues of the associated eigenvalue probl.
Author: Mikhail Shashkov
Publisher: CRC Press
ISBN: 9780849373756
Category : Mathematics
Languages : en
Pages : 384
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Book Description
This new book deals with the construction of finite-difference (FD) algorithms for three main types of equations: elliptic equations, heat equations, and gas dynamic equations in Lagrangian form. These methods can be applied to domains of arbitrary shapes. The construction of FD algorithms for all types of equations is done on the basis of the support-operators method (SOM). This method constructs the FD analogs of main invariant differential operators of first order such as the divergence, the gradient, and the curl. This book is unique because it is the first book not in Russian to present the support-operators ideas. Conservative Finite-Difference Methods on General Grids is completely self-contained, presenting all the background material necessary for understanding. The book provides the tools needed by scientists and engineers to solve a wide range of practical engineering problems. An abundance of tables and graphs support and explain methods. The book details all algorithms needed for implementation. A 3.5" IBM compatible computer diskette with the main algorithms in FORTRAN accompanies text for easy use.
Author: Zi-Cai Li
Publisher: World Scientific
ISBN: 9789810202927
Category : Mathematics
Languages : en
Pages : 286
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Book Description
This book presents two kinds of numerical methods for solving elliptic boundary value problems with singularities. Part I gives the boundary methods which use analytic and singular expansions, and Part II the nonconforming methods combining finite element methods (FEM) (or finite difference methods (FDM)) and singular (or analytic) expansions. The advantage of these methods over the standard FEM and FDM is that they can cope with complicated geometrical boundaries and boundary conditions as well as singularity. Therefore, accurate numerical solutions near singularities can be obtained. The description of methods, error bounds, stability analysis and numerical experiments are provided for the typical problems with angular, interface and infinity singularities. However, the approximate techniques and coupling strategy given can be applied to solving other PDE and engineering problems with singularities as well. This book is derived from the author's Ph. D. thesis which won the 1987 best doctoral dissertation award given by the Canadian Applied Mathematics Society.
Author: Qiwei Feng
Publisher:
ISBN:
Category : Finite differences
Languages : en
Pages : 0
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Book Description
Interface problems arise in many applications such as modeling of underground waste disposal, oil reservoirs, composite materials, and many others. The coefficient $a$, the source term $f$, the solution $u$ and the flux $a\nabla u\cdot \vec{n}$ are possibly discontinuous across the interface curve $\Gamma$ in such problems. In realistic problems, the coefficient $a$ may have large jumps across the interface curve, or it can be highly oscillatory across the whole domain. This leads to accuracy deterioration and huge condition numbers of resulting linear systems. In order to obtain reasonable numerical solutions, higher order numerical schemes are desirable. In Chapter 2 we propose a sixth order compact 9-point finite difference method (FDM) on uniform Cartesian grids, for Poisson interface problems with singular sources in a rectangular domain. The matrix $A$ in the resulting linear system $Ax=b$, following from the proposed compact 9-point scheme, is independent of any source terms $f$, jump conditions, and interface curves $\Gamma$. We prove the sixth order convergence rate for the proposed compact 9-point scheme using the discrete maximum principle. Our numerical experiments confirm the sixth order of accuracy of the proposed compact 9-point scheme. This chapter has been published in \emph{Computers and Mathematics with Applications} in 2021. In Chapter 3, elliptic interface problems with discontinuous and high-contrast piecewise smooth coefficients in a rectangle are considered. We propose a high order compact 9-point FDM and a high order local calculation for approximation of the solution $u$ and the gradient $\nabla u$ respectively. The scheme is developed on uniform Cartesian grids, avoiding the transformation into local coordinates. We also numerically verify the sign conditions of our proposed compact 9-point scheme and prove the fourth order convergence rate by the discrete maximum principle. Our numerical experiments confirm the fourth order accuracy for the numerically approximated solution $u$ in both $l2$ and $l{\infty}$ norms, and the fourth/third order accuracy for the numerically approximated gradient $\big((uh)x,(uh)y\big)$ in the $l2$/$l{\infty}$ norm. This chapter has been published in \emph{Applied Mathematics and Computation} in 2022. In Chapter 4, we propose an efficient and flexible way to achieve the implementation of a hybrid FDM in uniform Cartesian meshes for elliptic interface problems with discontinuous and high-contrast piecewise smooth coefficients in a rectangular domain. The scheme utilizes a 9-point compact stencil with a sixth order accuracy for interior regular points and 13-point stencil with a fifth order accuracy for interior irregular points. Near the boundary, the stencil is reduced to six points and near the domain corners - to four points, and the corresponding discretization has a sixth order of accuracy on uniform Cartesian meshes, for various boundary conditions (Dirichlet, Neumann and Robin). Our numerical experiments confirm the flexibility and the accuracy order in $l2$ and $l{\infty}$ norms. In Chapter 5, we present a sixth order compact FDM on uniform Cartesian meshes for the Helmholtz equation with singular sources, and any possible combination of boundary conditions (Dirichlet, Neumann, and impedance) in a rectangular domain. To reduce the pollution effect, we propose a new pollution minimization strategy that is based on the average truncation error of plane waves. Our numerical experiments demonstrate the superiority of the proposed compact finite difference scheme with reduced pollution effect, as compared to several state-of-the-art finite difference schemes in the literature, particularly in the critical pre-asymptotic region where $\textsf{k}h$ is near $1$ with $\textsf{k}$ being the wavenumber and $h$ the mesh size. This chapter has been submitted in \emph{SIAM Journal on Scientific Computing}. In Chapter 6, we propose a sixth order compact 9-point FDM on uniform Cartesian meshes for elliptic interface problems with particular intersecting interfaces and four discontinuous constant coefficients in a square domain, where the solution is smooth enough, and interface curves are horizontal and vertical straight lines. The formulas of proposed sixth order compact 9-point finite difference scheme are constructed explicitly for all grid points (regular points, interface points, and the intersection point). We prove the order $6$ convergence of our proposed compact 9-point scheme by the discrete maximum principle. Our numerical experiments confirm the flexibility and the sixth order accuracy in $l2$ and $l{\infty}$ norms of our proposed compact 9-point scheme.
Author: Bernd Heinrich
Publisher: Mathematical Research
ISBN: 9783112720882
Category : Mathematics
Languages : en
Pages : 0
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Book Description
No detailed description available for "Finite Difference Methods on Irregular Networks".
Author: Lourenco Beirao da Veiga
Publisher: Springer
ISBN: 3319026631
Category : Mathematics
Languages : en
Pages : 399
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Book Description
This book describes the theoretical and computational aspects of the mimetic finite difference method for a wide class of multidimensional elliptic problems, which includes diffusion, advection-diffusion, Stokes, elasticity, magnetostatics and plate bending problems. The modern mimetic discretization technology developed in part by the Authors allows one to solve these equations on unstructured polygonal, polyhedral and generalized polyhedral meshes. The book provides a practical guide for those scientists and engineers that are interested in the computational properties of the mimetic finite difference method such as the accuracy, stability, robustness, and efficiency. Many examples are provided to help the reader to understand and implement this method. This monograph also provides the essential background material and describes basic mathematical tools required to develop further the mimetic discretization technology and to extend it to various applications.
Author: Zi Cai Li
Publisher: Springer Science & Business Media
ISBN: 1461333385
Category : Mathematics
Languages : en
Pages : 488
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Book Description
In this book the author sets out to answer two important questions: 1. Which numerical methods may be combined together? 2. How can different numerical methods be matched together? In doing so the author presents a number of useful combinations, for instance, the combination of various FEMs, the combinations of FEM-FDM, REM-FEM, RGM-FDM, etc. The combined methods have many advantages over single methods: high accuracy of solutions, less CPU time, less computer storage, easy coupling with singularities as well as the complicated boundary conditions. Since coupling techniques are essential to combinations, various matching strategies among different methods are carefully discussed. The author provides the matching rules so that optimal convergence, even superconvergence, and optimal stability can be achieved, and also warns of the matching pitfalls to avoid. Audience: The book is intended for both mathematicians and engineers and may be used as text for advanced students.
Author: Jan Nordström
Publisher:
ISBN:
Category :
Languages : en
Pages : 38
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Book Description
Boundary and interface conditions are derived for high order finite difference methods applied to multidimensional linear problems in curvilinear coordinates. The boundary and interface conditions lead to conservative schemes and strict and strong stability provided that certain metric conditions are met.
Author: A.A. Samarskij
Publisher: Birkhäuser
ISBN: 3034892721
Category : Mathematics
Languages : en
Pages : 273
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Book Description
The finite-difference solution of mathematical-physics differential equations is carried out in two stages: 1) the writing of the difference scheme (a differ ence approximation to the differential equation on a grid), 2) the computer solution of the difference equations, which are written in the form of a high order system of linear algebraic equations of special form (ill-conditioned, band-structured). Application of general linear algebra methods is not always appropriate for such systems because of the need to store a large volume of information, as well as because of the large amount of work required by these methods. For the solution of difference equations, special methods have been developed which, in one way or another, take into account special features of the problem, and which allow the solution to be found using less work than via the general methods. This work is an extension of the book Difference M ethod3 for the Solution of Elliptic Equation3 by A. A. Samarskii and V. B. Andreev which considered a whole set of questions connected with difference approximations, the con struction of difference operators, and estimation of the ~onvergence rate of difference schemes for typical elliptic boundary-value problems. Here we consider only solution methods for difference equations. The book in fact consists of two volumes.
