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Numerical Solution of Differential Equations by Zhilin Li, Zhonghua Qiao, Tao Tang
(Introduction to Finite Difference and Finite Element Methods)

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Numerical Solution of Differential Equations (Introduction to Finite Difference and Finite Element Methods) written by Zhilin Li , North Carolina State University, USA, Zhonghua Qiao , Hong Kong Polytechnic University, China and Tao Tang , Southern University of Science and Technology, China. Publisher: Cambridge University Press, Year: 2018.
This introduction to finite difference and finite element methods is aimed at advanced undergraduate and graduate students who need to solve differential equations. The prerequisites are few (basic calculus, linear algebra, and ordinary and partial differential equations) and so the book will be accessible and useful to readers from a range of disciplines across science and engineering.
Part I begins with finite difference methods. Finite element methods are then introduced in Part II. In each part, the authors begin with a comprehensive discussion of one-dimensional problems, before proceeding to consider two or higher dimensions. An emphasis is placed on numerical algorithms, related mathematical theory, and essential details in the implementation, while some useful packages are also introduced. The authors also provide well-tested Matlab® codes, all available online.

Numerical Solution of Differential Equations (Introduction to Finite Difference and Finite Element Methods) written by Zhilin Li, Zhonghua Qiao, Tao Tang cover the following topics.

  • 1. Introduction
    1.1 Boundary Value Problems of Differential Equations
    1.2 Further Reading

  • 2. Finite Difference Methods for 1D Boundary Value Problems
    2.1 A Simple Example of a Finite Difference Method
    2.2 Fundamentals of Finite Difference Methods
    2.3 Deriving FD Formulas Using the Method of Undetermined Coefficients
    2.4 Consistency, Stability, Convergence, and Error Estimates of FD Methods
    2.5 FD Methods for 1D Self-adjoint BVPs
    2.6 FD Methods for General 1D BVPs
    2.7 The Ghost Point Method for Boundary Conditions Involving Derivatives
    2.8 An Example of a Nonlinear BVP
    2.9 The Grid Refinement Analysis Technique
    2.10 * 1D IIM for Discontinuous Coefficients

  • 3. Finite Difference Methods for 2D Elliptic PDEs
    3.1 Boundary and Compatibility Conditions
    3.2 The Central Finite Difference Method for Poisson Equations
    3.3 The Maximum Principle and Error Analysis
    3.4 Finite Difference Methods for General Second-order Elliptic PDEs
    3.5 Solving the Resulting Linear System of Algebraic Equations
    3.6 A Fourth-order Compact FD Scheme for Poisson Equations
    3.7 A Finite Difference Method for Poisson Equations in Polar Coordinates
    3.8 Programming of 2D Finite Difference Methods

  • 4. FD Methods for Parabolic PDEs
    4.1 The Euler Methods
    4.2 The Method of Lines
    4.3 The Crank–Nicolson scheme
    4.4 Stability Analysis for Time-dependent Problems
    4.5 FD Methods and Analysis for 2D Parabolic Equations
    4.6 The ADI Method
    4.7 An Implicit–explicit Method for Diffusion and Advection Equations
    4.8 Solving Elliptic PDEs using Numerical Methods for Parabolic PDEs

  • 5. Finite Difference Methods for Hyperbolic PDEs
    5.1 Characteristics and Boundary Conditions
    5.2 Finite Difference Schemes
    5.3 The Modified PDE and Numerical Diffusion/Dispersion
    5.4 The Lax–Wendroff Scheme and Other FD methods
    5.5 Numerical Boundary Conditions
    5.6 Finite Difference Methods for Second-order Linear Hyperbolic PDEs
    5.7 Some Commonly Used FD Methods for Linear System of Hyperbolic PDEs
    5.8 Finite Difference Methods for Conservation Laws

  • 6. Finite Element Methods for 1D Boundary Value Problems
    6.1 The Galerkin FE Method for the 1D Model
    6.2 Different Mathematical Formulations for the 1D Model
    6.3 Key Components of the FE Method for the 1D Model
    6.4 Matlab Programming of the FE Method for the 1D Model Problem

  • 7. Theoretical Foundations of the Finite Element Method
    7.1 Functional Spaces
    7.2 Spaces for Integral Forms, L2 (?) and Lp(?)
    7.3 Sobolev Spaces and Weak Derivatives
    7.4 FE Analysis for 1D BVPs
    7.5 Error Analysis of the FE Method

  • 8. Issues of the FE Method in One Space Dimension
    8.1 Boundary Conditions
    8.2 The FE Method for Sturm–Liouville Problems
    8.3 High-order Elements
    8.4 A 1D Matlab FE Package
    8.5 The FE Method for Fourth-order BVPs in 1D
    8.6 The Lax–Milgram Lemma and the Existence of FE Solutions
    8.7 *1D IFEM for Discontinuous Coefficients

  • 9. The Finite Element Method for 2D Elliptic PDEs
    9.1 The Second Green’s Theorem and Integration by Parts in 2D
    9.2 Weak Form of Second-order Self-adjoint Elliptic PDEs
    9.3 Triangulation and Basis Functions
    9.4 Transforms, Shape Functions, and Quadrature Formulas
    9.5 Some Implementation Details
    9.6 Simplification of the FE Method for Poisson Equations
    9.7 Some FE Spaces in H1 (?) and H2 (?)
    9.8 The FE Method for Parabolic Problems

  • Appendix: Numerical Solutions of Initial Value Problems
    A.1 System of First-order ODEs of IVPs
    A.2 Well-posedness of an IVP
    A.3 Some Finite Difference Methods for Solving IVPs
    A.4 Solving IVPs Using Matlab ODE Suite

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