Numerical Methods for Partial Differential Equations

Numerical Methods for Partial Differential Equations

Finite Difference and Finite Volume Methods

1st Edition - December 1, 2015

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  • Author: Sandip Mazumder
  • eBook ISBN: 9780128035047
  • Paperback ISBN: 9780128498941

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Numerical Methods for Partial Differential Equations: Finite Difference and Finite Volume Methods focuses on two popular deterministic methods for solving partial differential equations (PDEs), namely finite difference and finite volume methods. The solution of PDEs can be very challenging, depending on the type of equation, the number of independent variables, the boundary, and initial conditions, and other factors. These two methods have been traditionally used to solve problems involving fluid flow. For practical reasons, the finite element method, used more often for solving problems in solid mechanics, and covered extensively in various other texts, has been excluded. The book is intended for beginning graduate students and early career professionals, although advanced undergraduate students may find it equally useful. The material is meant to serve as a prerequisite for students who might go on to take additional courses in computational mechanics, computational fluid dynamics, or computational electromagnetics. The notations, language, and technical jargon used in the book can be easily understood by scientists and engineers who may not have had graduate-level applied mathematics or computer science courses.

Key Features

  • Presents one of the few available resources that comprehensively describes and demonstrates the finite volume method for unstructured mesh used frequently by practicing code developers in industry
  • Includes step-by-step algorithms and code snippets in each chapter that enables the reader to make the transition from equations on the page to working codes
  • Includes 51 worked out examples that comprehensively demonstrate important mathematical steps, algorithms, and coding practices required to numerically solve PDEs, as well as how to interpret the results from both physical and mathematic perspectives


Graduate level Mechanical, Aerospace, Civil, Biomedical, and Chemical Engineering students; engineering professionals involved in areas such as computational mechanics, computational fluid dynamics, and computational electromagnetics

Table of Contents

    • Dedication
    • About the Author
    • Preface
    • List of Symbols
    • Chapter 1: Introduction to Numerical Methods for Solving Differential Equations
      • Abstract
      • 1.1. Role of Analysis
      • 1.2. Classification of PDEs
      • 1.3. Overview of methods for solving PDEs
      • 1.4. Overview of Mesh Types
      • 1.5. Verification and Validation
    • Chapter 2: The Finite Difference Method
      • Abstract
      • 2.1. Difference Approximations and Truncation Errors
      • 2.2. General Procedure for Deriving Difference Approximations
      • 2.3. Application of Boundary Conditions
      • 2.4. Assembly of Nodal Equations in Matrix Form
      • 2.5. Multidimensional Problems
      • 2.6. Higher-Order Approximations
      • 2.7. Difference Approximations in the Cylindrical Coordinate System
      • 2.8. Coordinate Transformation to Curvilinear Coordinates
    • Chapter 3: Solution to a System of Linear Algebraic Equations
      • Abstract
      • 3.1. Direct Solvers
      • 3.2. Iterative Solvers
      • 3.3. Overview of Other Methods
      • 3.4. Treatment of Nonlinear Sources
    • Chapter 4: Stability and Convergence of Iterative Solvers
      • Abstract
      • 4.1. Eigenvalues and Condition Number
      • 4.2. Stability
      • 4.3. Rate of Convergence
      • 4.4. Preconditioning
      • 4.5. Multigrid Method
    • Chapter 5: Treatment of the Time Derivative (Parabolic and Hyperbolic PDEs)
      • Abstract
      • 5.1. Steady-State Versus Time-Marching
      • 5.2. Parabolic Partial Differential Equations
      • 5.3. Hyperbolic Partial Differential Equations
      • 5.4. Higher Order Methods for Ordinary Differential Equations
      • 5.5. Method of Lines
    • Chapter 6: The Finite Volume Method (FVM)
      • Abstract
      • 6.1. Derivation of Finite Volume Equations
      • 6.2. Application of Boundary Conditions
      • 6.3. Flux Schemes for Advection–Diffusion
      • 6.4. Multidimensional Problems
      • 6.5. Two-Dimensional Axisymmetric Problems
      • 6.6. Finite Volume Method in Curvilinear Coordinates
      • 6.7. Summary of FDM and FVM
    • Chapter 7: Unstructured Finite Volume Method
      • Abstract
      • 7.1. Gauss Divergence Theorem and its Physical Significance
      • 7.2. Derivation of Finite Volume Equations on an Unstructured Mesh
      • 7.3. Processing and Storage of Geometric (Mesh) Information
      • 7.4. Treatment of Normal and Tangential Fluxes
      • 7.5. Boundary Condition Treatment
      • 7.6. Assembly and Solution of Discrete Equations
      • 7.7. Finite Volume Formulation for Advection–Diffusion Equation
    • Chapter 8: Miscellaneous Topics
      • Abstract
      • 8.1. Interpolation
      • 8.2. Numerical integration
      • 8.3. Newton–Raphson method for nonlinear equations
      • 8.4. Application of the Newton–Raphson method to solving nonlinear PDEs
      • 8.5. Solution of coupled PDEs
    • Appendix A: Useful Relationships in Matrix Algebra
    • Appendix B: Useful Relationships in Vector Calculus
    • Appendix C: Tensor Notations and Useful Relationships
    • Index

Product details

  • No. of pages: 484
  • Language: English
  • Copyright: © Academic Press 2015
  • Published: December 1, 2015
  • Imprint: Academic Press
  • eBook ISBN: 9780128035047
  • Paperback ISBN: 9780128498941

About the Author

Sandip Mazumder

Sandip Mazumder received his PhD from the Pennsylvania State University, and is currently Professor at The Ohio State University. His research in radiation has primarily involved developing efficient methods for solving the radiative transfer equation and coupling it to other modes of heat transfer for practical applications. Dr. Mazumder was employed at CFD Research Corporation for 7 years prior to joining Ohio State in 2004. He is the recipient of the McCarthy teaching award and the Lumley research award from the Ohio State College of Engineering, among other awards, and is a fellow of the ASME.

Affiliations and Expertise

Department of Mechanical and Aerospace Engineering, The Ohio State University, Columbus, OH, USA

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  • Braxton H. Mon May 14 2018

    Comprehensive Look at Numerical Methods

    This book succeeds at breaking down the derivations, implementation, and application of both the Finite Difference and Finite Volume Methods. It serves as a fantastic instructional and reference text for the novice numerical analyst, and sets the stage for advanced work in the computational field.