Introduction to Numerical Methods for Time Dependent Differential Equations

HEINZ-OTTO KREISS, PHD, is Professor Emeritus in the Department of Mathematics at the University of California, Los Angeles and is a renowned mathematician in the field of applied mathematics.OMAR EDUARDO ORTIZ, PHD, is Professor in the Department of Mathematics, Astronomy, and Physics at the National University of Córdoba, Argentina. Dr. Ortiz’s research interests include analytical and numerical methods for PDEs applied in physics.

• Preface xiiiAcknowledgments xvPART I ORDINARY DIFFERENTIAL EQUATIONS AND THEIR APPROXIMATIONS1 First Order Scalar Equations 31.1 Constant coefficient linear equations 31.1.1 Duhamel’s principle 81.1.2 Principle of frozen coefficients 101.2 Variable coefficient linear equations 101.2.1 The principle of superposition 101.2.2 Duhamel’s principle for variable coefficients 121.3 Perturbations and the concept of stability 131.4 Nonlinear equations: the possibility of blowup 171.5 The principle of linearization 202 The Method of Euler 232.1 The explicit Euler method 232.2 Stability of the explicit Euler method 252.3 Accuracy and truncation error 272.4 Discrete Duhamel’s principle and global error 282.5 General onestep methods. 312.6 How to test the correctness of a program 322.7 Extrapolation 343 Higher Order Methods 373.1 The secondorder Taylor method 373.2 Improved Euler’s method 393.3 Accuracy of the computed solution 403.4 RungeKutta methods 443.5 Regions of stability 483.6 Accuracy and truncation error 513.7 Difference approximations for unstable problems 524 The Implicit Euler Method 554.1 Stiff equations 554.2 The implicit Euler method 584.3 A simple variable step size strategy 635 Two Step and Multistep Methods 675.1 Multistep methods 675.2 The leapfrog method 685.3 Adams methods 725.4 Stability of multistep methods 746 Systems of Differential Equations 77PART II PARTIAL DIFFERENTIAL EQUATIONS AND THEIR APPROXIMATIONS7 Fourier Series and Interpolation 837.1 Fourier expansion 837.2 The L2norm and scalar product 897.3 Fourier interpolation 927.3.1 Scalar product and norm for 1periodic grid functions 938 1periodic Solutions of Time Dependent PDE... 958.1 Examples of equations with simple wave solutions 958.1.1 The oneway wave equation 958.1.2 The heat equation 968.1.3 The wave equation 978.2 Discussion of well posed problems for time dependent PDE... 988.2.1 First order equations 988.2.2 Second order (in space) equations 1008.2.3 General equation 1018.2.4 Stability against lower order terms and systems of equations 1029 Approximations of 1periodic Solutions of PDE 1059.1 Approximations of space derivatives 1059.1.1 Smoothness of the Fourier interpolant 1089.2 Differentiation of Periodic Functions 1099.3 The method of lines 1109.3.1 The oneway wave equation 1109.3.2 The heat equation 1139.3.3 The wave equation 1159.4 Time Discretizations and Stability Analysis 11610 Linear InitialBoundary Value Problems 11910.1 Well Posed InitialBoundary Value Problems 11910.1.1 The heat equation on a strip 12010.1.2 The oneway wave equation on a strip 12210.1.3 The wave equation on a strip 12410.2 The method of lines 12610.2.1 The heat equation 12610.2.2 Finite differences algebra 13010.2.3 General parabolic problem 13110.2.4 The oneway wave equation 13410.2.5 The wave equation 13511 Nonlinear Problems 13711.1 Initialvalue problems for ODE 13811.2 Existence theorems for nonlinear PDE 14111.3 A nonlinear example: Burgers’ equation 145A Auxiliary Material 149A.1 Some useful Taylor series 149A.2 The “O” notation 150A.3 The solution expansion 150B Solutions to Exercises 153References 171Index 173

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