Computational Science and Engineering

by Strang, Gilbert

• ISBN13: 9780961408817
• ISBN10: 0961408812
• Format: Hardcover
• Copyright: 2007-10-01
• Publisher: Wellesley Cambridge Pr

Summary

Encompasses the full range of computational science and engineering from modelling to solution, both analytical and numerical. It develops a framework for the equations and numerical methods of applied mathematics. Gilbert Strang has taught this material to thousands of engineers and scientists (and many more on MIT's OpenCourseWare 18.085-6). His experience is seen in his clear explanations, wide range of examples, and teaching method. The book is solution-based and not formula-based: it integrates analysis and algorithms and MATLAB codes to explain each topic as effectively as possible. The topics include applied linear algebra and fast solvers, differential equations with finite differences and finite elements, Fourier analysis and optimization. This book also serves as a reference for the whole community of computational scientists and engineers. Supporting resources, including MATLAB codes, problem solutions and video lectures from Gilbert Strang's 18.085 courses at MIT, are provided at math.mit.edu/cse.

Table of Contents

Applied Linear Algebra

Four special matrices

Differences, derivatives, and boundary conditions

Elimination leads to K = LDL^T

Inverses and delta functions

Eigenvalues and eigenvectors

Positive definite matrices

Numerical linear algebra: LU, QR, SVD

Best basis from the SVD

A Framework for Applied Mathematics

Equilibrium and the stiffness matrix

Oscillation by Newton's law

Least squares for rectangular matrices

Graph models and Kirchhoff's laws

Networks and transfer functions

Nonlinear problems

Structures in equilibrium

Covariances and recursive least squares

Graph cuts and gene clustering

Boundary Value Problems

Differential equations of equilibrium

Cubic splines and fourth order equations

Gradient and divergence

Laplace's equation

Finite differences and fast Poisson solvers

The finite element method

Elasticity and solid mechanics

Fourier Series and Integrals

Fourier series for periodic functions

Chebyshev, Legendre, and Bessel

The discrete Fourier transform and the FFT

Convolution and signal processing

Fourier integrals

Deconvolution and integral equations

Wavelets and signal processing

Analytic Functions

Taylor series and complex integration

Famous functions and great theorems

The Laplace transform and z-transform

Spectral methods of exponential accuracy

Initial Value Problems

Introduction

Finite difference methods for ODEs

Accuracy and stability for u_t = c u_x

The wave equation and staggered leapfrog

Diffusion, convection, and finance

Nonlinear flow and conservation laws

Fluid mechanics and Navier-Stokes

Level sets and fast marching

Solving Large Systems

Elimination with reordering

Iterative methods

Multigrid methods

Conjugate gradients and Krylov subspaces

Optimization and Minimum Principles

Two fundamental examples

Regularized least squares

Calculus of variations

Errors in projections and eigenvalues

The Saddle Point Stokes problem

Linear programming and duality

Adjoint methods in design

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