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Part I: Ordinary Differential Equations and Their Approximations
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Part I: Ordinary Differential Equations and Their Approximations
by Omar Eduardo Ortiz, Heinz-Otto Kreiss
Introduction to Numerical Methods for Time Dependent Differential Equations
Cover
Half Title page
Title page
Copyright page
Dedication
Preface
Acknowledgements
Part I: Ordinary Differential Equations and Their Approximations
Chapter 1: First-Order Scalar Equations
1.1 Constant coefficient linear equations
1.2 Variable coefficient linear equations
1.3 Perturbations and the concept of stability
1.4 Nonlinear equations: the possibility of blow-up
1.5 Principle of linearization
Chapter 2: Method of Euler
2.1 Explicit Euler method
2.2 Stability of the explicit Euler method
2.3 Accuracy and truncation error
2.4 Discrete Duhamel’s principle and global error
2.5 General one-step methods
2.6 How to test the correctness of a program
2.7 Extrapolation
Chapter 3: Higher-Order Methods
3.1 Second-order Taylor method
3.2 Improved Euler’s method
3.3 Accuracy of the solution computed
3.4 Runge-Kutta methods
3.5 Regions of stability
3.6 Accuracy and truncation error
3.7 Difference approximations for unstable problems
Chapter 4: Implicit Euler Method
4.1 Stiff equations
4.2 Implicit Euler method
4.3 Simple variable-step-size strategy
Chapter 5: Two-Step and Multistep Methods
5.1 Multistep methods
5.2 Leapfrog method
5.3 Adams methods
5.4 Stability of multistep methods
Chapter 6: Systems of Differential Equations
Part II: Partial Differential Equations and Their Approximations
Chapter 7: Fourier Series and Interpolation
7.1 Fourier expansion
7.2 L2-norm and scalar product
7.3 Fourier interpolation
Chapter 8: 1-Periodic Solutions of time Dependent Partial Differential Equations with Constant Coefficients
8.1 Examples of equations with simple wave solutions
8.2 Discussion of well posed problems for time dependent partial differential equations with constant coefficients and with 1-periodic boundary conditions
Chapter 9: Approximations of 1-Periodic Solutions of Partial Differential Equations
9.1 Approximations of space derivatives
9.2 Differentiation of Periodic Functions
9.3 Method of lines
9.4 Time Discretizations and Stability Analysis
Chapter 10: Linear Initial Boundary Value Problems
10.1 Well-Posed Initial Boundary Value Problems
10.2 Method of lines
Chapter 11: Nonlinear Problems
11.1 Initial value problems for ordinary differential equations
11.2 Existence theorems for nonlinear partial differential equations
11.3 Nonlinear example: Burgers’ equation
Appendix A: Auxiliary Material
A.1 Some useful Taylor series
A.2 “O” notation
A.3 Solution expansion
Appendix B: Solutions to Exercises
References
Index
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Chapter 1: First-Order Scalar Equations
PART I
ORDINARY DIFFERENTIAL EQUATIONS AND THEIR APPROXIMATIONS
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