Introduction to Numerical Methods for Time Dependent Differential Equations

[BOOK DESCRIPTION]

Introduces both the fundamentals of time dependent differential equations and their numerical solutions Introduction to Numerical Methods for Time Dependent Differential Equations delves into the underlying mathematical theory needed to solve time dependent differential equations numerically. Written as a self-contained introduction, the book is divided into two parts to emphasize both ordinary differential equations (ODEs) and partial differential equations (PDEs). Beginning with ODEs and their approximations, the authors provide a crucial presentation of fundamental notions, such as the theory of scalar equations, finite difference approximations, and the Explicit Euler method. Next, a discussion on higher order approximations, implicit methods, multistep methods, Fourier interpolation, PDEs in one space dimension as well as their related systems is provided.Introduction to Numerical Methods for Time Dependent Differential Equations features: * A step-by-step discussion of the procedures needed to prove the stability of difference approximations * Multiple exercises throughout with select answers, providing readers with a practical guide to understanding the approximations of differential equations * A simplified approach in a one space dimension * Analytical theory for difference approximations that is particularly useful to clarify procedures Introduction to Numerical Methods for Time Dependent Differential Equations is an excellent textbook for upper-undergraduate courses in applied mathematics, engineering, and physics as well as a useful reference for physical scientists, engineers, numerical analysts, and mathematical modelers who use numerical experiments to test designs or predict and investigate phenomena from many disciplines.

[TABLE OF CONTENTS]

Preface                                            xi
Acknowledgments                                    xiii
Part I Ordinary Differential Equations And
Their Approximations
  1 First-Order Scalar Equations                   3   (20)
    1.1 Constant coefficient linear equations      3   (7)
      1.1.1 Duhamel's principle                    8   (1)
      1.1.2 Principle of frozen coefficients       9   (1)
    1.2 Variable coefficient linear equations      10  (4)
      1.2.1 Principle of superposition             10  (2)
      1.2.2 Duhamel's principle for variable       12  (2)
      coefficients
    1.3 Perturbations and the concept of           14  (4)
    stability
    1.4 Nonlinear equations: the possibility of    18  (2)
    blow-up
    1.5 Principle of linearization                 20  (3)
  2 Method of Euler                                23  (14)
    2.1 Explicit Euler method                      23  (3)
    2.2 Stability of the explicit Euler method     26  (1)
    2.3 Accuracy and truncation error              27  (2)
    2.4 Discrete Duhamel's principle and global    29  (3)
    error
    2.5 General one-step methods                   32  (1)
    2.6 How to test the correctness of a program   32  (3)
    2.7 Extrapolation                              35  (2)
  3 Higher-Order Methods                           37  (18)
    3.1 Second-order Taylor method                 37  (2)
    3.2 Improved Euler's method                    39  (1)
    3.3 Accuracy of the solution computed          40  (4)
    3.4 Runge-Kutta methods                        44  (4)
    3.5 Regions of stability                       48  (3)
    3.6 Accuracy and truncation error              51  (1)
    3.7 Difference approximations for unstable     52  (3)
    problems
  4 Implicit Euler Method                          55  (10)
    4.1 Stiff equations                            55  (3)
    4.2 Implicit Euler method                      58  (5)
    4.3 Simple variable-step-size strategy         63  (2)
  5 Two-Step and Multistep Methods                 65  (10)
    5.1 Multistep methods                          65  (1)
    5.2 Leapfrog method                            66  (4)
    5.3 Adams methods                              70  (1)
    5.4 Stability of multistep methods             71  (4)
  6 Systems of Differential Equations              75  (6)
Part II Partial Differential Equations And
Their Approximations
  7 Fourier Series and Interpolation               81  (12)
    7.1 Fourier expansion                          81  (6)
    7.2 L2-norm and scalar product                 87  (3)
    7.3 Fourier interpolation                      90  (3)
      7.3.1 Scalar product and norm for            91  (2)
      1-periodic grid functions
  8 1-Periodic Solutions of Time Dependent         93  (12)
  Partial Differential Equations with Constant
  Coefficients
    8.1 Examples of equations with simple wave     93  (3)
    solutions
      8.1.1 One-way wave equation                  93  (1)
      8.1.2 Heat equation                          94  (1)
      8.1.3 Wave equation                          95  (1)
    8.2 Discussion of well posed problems for      96  (9)
    time dependent partial differential
    equations with constant coefficients and
    with 1-periodic boundary conditions
      8.2.1 First-order equations                  96  (2)
      8.2.2 Second-order (in space) equations      98  (1)
      8.2.3 General equation                       99  (1)
      8.2.4 Stability against lower-order terms    100 (5)
      and systems of equations
  9 Approximations of 1-Periodic Solutions of      105 (14)
  Partial Differential Equations
    9.1 Approximations of space derivatives        105 (4)
      9.1.1 Smoothness of the Fourier              108 (1)
      interpolant
    9.2 Differentiation of Periodic Functions      109 (1)
    9.3 Method of lines                            110 (5)
      9.3.1 One-way wave equation                  110 (3)
      9.3.2 Heat equation                          113 (1)
      9.3.3 Wave equation                          114 (1)
    9.4 Time Discretizations and Stability         115 (4)
    Analysis
  10 Linear Initial Boundary Value Problems        119 (18)
    10.1 Well-Posed Initial Boundary Value         119 (7)
    Problems
      10.1.1 Heat equation on a strip              120 (2)
      10.1.2 One-way wave equation on a strip      122 (2)
      10.1.3 Wave equation on a strip              124 (2)
    10.2 Method of lines                           126 (11)
      10.2.1 Heat equation                         126 (4)
      10.2.2 Finite-differences algebra            130 (1)
      10.2.3 General parabolic problem             131 (3)
      10.2.4 One-way wave equation                 134 (1)
      10.2.5 Wave equation                         135 (2)
  11 Nonlinear Problems                            137 (12)
    11.1 Initial value problems for ordinary       138 (3)
    differential equttons
    11.2 Existence theorems for nonlinear          141 (4)
    partial differential equations
    11.3 Nonlinear example: Burgers' equation      145 (4)
A Auxiliary Material                               149 (6)
  A.1 Some useful Taylor series                    149 (1)
  A.2 "O" notation                                 150 (1)
  A.3 Solution expansion                           150 (5)
B Solutions to Exercises                           155 (18)
References                                         173 (2)
Index                                              175