Stefania Tomasiello (University of Basilicata, Italy)

Copyright: © 2012
|Pages: 31

DOI: 10.4018/978-1-61350-116-0.ch014

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TopA variety of numerical methods are available for solving problems in science and engineering, when analytical solutions (so-called closed-form solutions) cannot be achieved. Numerical methods are part of the field called *scientific computing*, which can be defined as (Golub and Ortega, 1992, p. 2):

“The collection of tools, techniques, and theories required to solve on a computer mathematical models of problems in science and engineering”.

Mathematical models very often take the form of differential equations, with initial and/or boundary conditions. The variables of the problem may depend on time and/or *n-*space coordinates respectively (in this latter case the problem is called *n-*dimensional, where *n=1,2,3* usually). For example, a boundary condition can represent restraints i.e. the supports of the multi-span beam (see the first example in the final section) and an initial condition can be the initial heat profile in the time-dependent problem of heat transfer (see the second example in the final section).

Numerical computations are affected by several types of error, *rounding errors* are due to machine precision. *Discretization errors* are created when the “continuous” problem domain is substituted by a “discrete” one. The *convergence error* is due to the iterative nature of some numerical methods, where a finite sequence of approximations to a solution is generated. An acceptable degree of error depends on the problem being solved: for many problems in industry or engineering three digit accuracy is quite acceptable.

Discretization is the core of almost all the numerical methods (including the methods discussed in this chapter), by introducing a set of points into the problem or computational spatial domain (i.e. where the solution will be computed). These points, often equally spaced, can have different names, e.g. *grid points,* in the Finite Difference Method (FDM), *collocation points* in the collocation methods or *nodes* in the Finite Element Method (FEM). Choosing the points in the computational domain is achieved by constructing the approximate solution of the values at these points. This has the benefit of converting the differential equations into an algebraic system of equations that can be easily solved on a computer.

Replacing the differential equations with an algebraic system of equations can be done by approximating the derivatives of the unknown function (the solution of the problem) by formulas based on the values of the function itself at discrete points and is generally achieved in a systematic way (e.g in FDM) or by using polynomials which interpolate the function at the given points (e.g. FEM or collocation methods). How these interpolating polynomials are chosen and used is the main difference between some numerical methods. In collocation methods a finite-dimensional space of candidate solutions (usually, polynomials up to a certain degree) are chosen, so that the solution satisfies the given equation at the collocation points. The FEM approximates the solution as a linear combination of piecewise functions (usually, polynomials), that are nonzero on small sub-domains. In fact, the basic idea of FEM is to divide the spatial domain of the problem into smaller parts (or sub-domains) called *finite elements* or *elements,* connected by points called *nodes*, giving a topological map which is called a *mesh*, whereas the process of making the mesh is called *mesh generation*. The main advantage of the FEM is its ability to reproduce non-smooth solutions, in conjunction with the ability to handle arbitrarily shaped domains. For example, stress analysis is an important process in engineering that requires the solution of a system of partial differential equations that are very difficult to solve by analytical methods except for very simple shapes, such as rectangles; however, engineering problems seldom involve such simple shapes. For linear problems, the solution is determined by solving a system of linear equations, for a certain number of unknowns, which are the nodal values of the problem function. With regard to engineering problems, in order to obtain a reasonably accurate solution, thousands of nodes are usually needed. Generally, the accuracy of the solution improves as the number of elements (and nodes) increases, but the computing time (and hence the cost) also rises.

Boundary Conditions: The conditions that are satisfied on the boundaries of the spatial domain of the problem. The governing differential equation with the boundary conditions represents a boundary value problem (BVP)

Nodes: Points in space where finite elements are connected and where the unknowns of the algebraic equation system are computed.

Grid Points: Points are selected in the problem spatial domain where the numerical solution is computed.

FDM: The Finite Difference Method computes approximate values for the solution at grid points. To compute these values, derivatives are replaced by divided differences. This method can be useful for solving heat transfer problems, fluid mechanics problems and works well for two-dimensional regions with boundaries parallel to coordinates axes.

Meshfree Methods: Numerical method which compute the approximate solution without a predefined mesh.

Initial Conditions: For time dependent problems, the conditions that must be satisfied at the initial/start time. The governing equations with the initial conditions represent the initial value problem (IVP).

Collocation Points: see collocation methods

Collocation Methods: The approximate solution is produced by means of the values it assumes in some locations, called collocation points, where the governing differential equation is satisfied.

FEM: The Finite Element Method, computes the solution by piecewise functions being computed in smaller parts of the problem domain, called finite elements. A set of algebric equations are established and solved. It is employed extensively in the analysis of solids, structures, heat transfer and fluids. It can handle arbitrarily shaped domains.

MeSH: Topological map of the finite element spatial domain as discrete points rather than a continuous field.

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Table of Contents

Preface

Joanna Leng, Wes Sharrock

Chapter 1

Gabriele Jost, Alice E. Koniges

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Chapter 2

Ivan Girotto, Robert M. Farber

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Chapter 4

High-Performance Computing for Theoretical Study of Nanoscale and Molecular Interconnects
(pages 78-97)

Rasit O. Topaloglu, Swati R. Manjari, Saroj K. Nayak

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Chapter 6

Pragmatic Software Engineering for Computational Science
(pages 119-149)

David Worth, Chris Greenough, Shawn Chin

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Chapter 7

A Framework for Testing Code in Computational Applications
(pages 150-176)

Diane Kelly, Daniel Hook, Rebecca Sanders

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Chapter 8

Judith Segal, Chris Morris

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Chapter 9

Opportunities and Challenges in Porting a Parallel Code from a Tightly-Coupled System to the Distributed EU Grid, Enabling Grids for E-sciencE
(pages 197-217)

Fumie Costen, Akos Balasko

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Chapter 10

Development of an Efficient and Secure Mobile Communication System with New Future Directions
(pages 219-238)

Abid Yahya, Farid Ghani, R. Badlishah Ahmad, Mostafijur Rahman, Aini Syuhada, Othman Sidek, M. F. M. Salleh

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Chapter 13

C. T. J. Dodson

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Chapter 14

Stefania Tomasiello

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Chapter 15

Mesh Morphing and Smoothing by Means of Radial Basis Functions (RBF): A Practical Example Using Fluent and RBF Morph
(pages 347-380)

Marco Evangelos Biancolini

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Chapter 16

Visualization: Future Technology and Practices for Computational Science and Engineering
(pages 381-413)

Joanna Leng, Theresa-Marie Rhyne, Wes Sharrock

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Chapter 17

Visualizing Indicators of Debt Crises in a Lower Dimension: A Self-Organizing Maps Approach
(pages 414-431)

Peter Sarlin

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Chapter 18

Improving Computational Models and Practices: Scenario Testing and Forecasting the Spread of Infectious Disease
(pages 432-455)

Iain Barrass, Joanna Leng

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Chapter 19

Eldon R. Rene, Sung Joo Kim, Dae Hee Lee, Woo Bong Je, Mirian Estefanía López, Hung Suck Park

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Chapter 21

Integrating Data Management and Collaborative Sharing with Computational Science Research Processes
(pages 506-538)

Kerstin Kleese van Dam, Mark James, Andrew M. Walker

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Chapter 22

Security and Trust in a Global Research Infrastructure
(pages 539-566)

Jens Jensen, David L. Groep

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Chapter 23

CSE as Epistemic Technologies: Computer Modeling and Disciplinary Difference in the Humanities
(pages 567-586)

Matt Ratto

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Chapter 24

Science Communication with Dinosaurs
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Phillip L. Manning, Peter L. Falkingham

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About the Contributors

Index