The Computable Differential Equation Lecture ... - Bruce E. Shapiro
The Computable Differential Equation Lecture ... - Bruce E. Shapiro The Computable Differential Equation Lecture ... - Bruce E. Shapiro
iv CONTENTS 5.3 Traditional Runge-Kutta Methods . . . . . . . . . . . . . . . . . . . 95 5.4 General Form of Runge-Kutta Methods . . . . . . . . . . . . . . . . 99 5.5 Order Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 5.6 Step Size Control for ERK Methods . . . . . . . . . . . . . . . . . . 104 5.7 Implicit Runge-Kutta Methods . . . . . . . . . . . . . . . . . . . . . 106 5.8 IRK Methods based on Collocation . . . . . . . . . . . . . . . . . . . 109 5.9 Páde Approximants and A-Stability . . . . . . . . . . . . . . . . . . 112 6 Linear Multistep Methods 117 6.1 Multistep Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 6.2 The Root Condition for Linear Multistep Methods . . . . . . . . . . 119 6.3 Backward Difference Formula . . . . . . . . . . . . . . . . . . . . . . 121 6.4 Adams Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 6.5 BDF Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 6.6 Nyström and Milne Methods . . . . . . . . . . . . . . . . . . . . . . 128 6.7 Other Types of Multistep Methods . . . . . . . . . . . . . . . . . . . 130 6.8 Prediction-Correction (P(EC) n E) Techniques . . . . . . . . . . . . . 133 7 Delay Differential Equations 135 7.1 Basic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 7.2 Method Of Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 7.3 Numerical Implementation of Method of Steps . . . . . . . . . . . . 143 7.4 Runge-Kutta Methods for Delay Differential Equations . . . . . . . . 147 7.5 Continuous Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 8 Boundary Value Problems 153 8.1 Shooting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 8.2 Basic Theory of Boundary Value Problems . . . . . . . . . . . . . . 156 8.3 Shooting Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 8.4 One-step Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 9 Differential Algebraic Equations 167 9.1 Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 9.2 Linear Differential Algebraic Equations with Constant Coeffcients . 170 9.3 General Linear Differential Algebraic Equations . . . . . . . . . . . . 173 9.4 Hessenberg Forms for Linear and Nonlinear DAEs . . . . . . . . . . 174 9.5 Implementation: A Detailed Example . . . . . . . . . . . . . . . . . 177 9.6 BDF Methods for DAEs . . . . . . . . . . . . . . . . . . . . . . . . . 184 9.7 Runge-Kutta Methods for DAEs . . . . . . . . . . . . . . . . . . . . 187 10 Appendix on Analytic Methods (Math 280 in a Nutshell) 193 Bibliography 199 Math 582B, Spring 2007 California State University Northridge c○2007, B.E.Shapiro Last revised: May 23, 2007
CONTENTS v Timeline on Computable Differential Equations 1660s Newton circulates notes on limit theory of Calculus 1674 Leibnitz, infinitesimal theory of Calculus 1691 Raphson, root finding algorithm 17?? Newton invents Simpson’s Method 1739 Newton Publishes 1740 Simpson invents Newton’s Method 17?? Euler’s Method 1755,1768 Euler publishes Calculus texts 1883 Bashforth, Adams 1885 Schwartz, Method of Successive Iterations 1887 Peano’s theory 1890 Picard; Lindeloff 1895 Runge 1901 Kutta; Heun 1926 Moulton; Milne 1951 UNIVAC - first commercial computer 1957 First FORTRAN compiler 1963++ Butcher, Theory of RK 1963++ Driver, Theory of Delay Equations 1968-1977 Development of Macsyma 1969 APRAnet goes online 1971 Gear, BDF Methods for ODE’s and DAE’s 1974 GEAR (Hindmarsh) 1980 LSODE/LSODI (Hindmarsh, Petzold) 1982 - Present DASSL Family 1983 ODEPACK 1985 Netlib goes online 1986 Petzold, RK Methods for DAE’s 1986 GMRES 1988 Numerical Recipes Published 1988 Mathematica 1.0 1989 VODE 1994-1996 CVODE/PVODE family 1999 IDA (parallel version of CVODE/PVODE) 2005 Sundials c○2007, B.E.Shapiro Last revised: May 23, 2007 Math 582B, Spring 2007 California State University Northridge
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CONTENTS<br />
v<br />
Timeline on <strong>Computable</strong> <strong>Differential</strong> <strong>Equation</strong>s<br />
1660s Newton circulates notes on limit theory of Calculus<br />
1674 Leibnitz, infinitesimal theory of Calculus<br />
1691 Raphson, root finding algorithm<br />
17?? Newton invents Simpson’s Method<br />
1739 Newton Publishes<br />
1740 Simpson invents Newton’s Method<br />
17?? Euler’s Method<br />
1755,1768 Euler publishes Calculus texts<br />
1883 Bashforth, Adams<br />
1885 Schwartz, Method of Successive Iterations<br />
1887 Peano’s theory<br />
1890 Picard; Lindeloff<br />
1895 Runge<br />
1901 Kutta; Heun<br />
1926 Moulton; Milne<br />
1951 UNIVAC - first commercial computer<br />
1957 First FORTRAN compiler<br />
1963++ Butcher, <strong>The</strong>ory of RK<br />
1963++ Driver, <strong>The</strong>ory of Delay <strong>Equation</strong>s<br />
1968-1977 Development of Macsyma<br />
1969 APRAnet goes online<br />
1971 Gear, BDF Methods for ODE’s and DAE’s<br />
1974 GEAR (Hindmarsh)<br />
1980 LSODE/LSODI (Hindmarsh, Petzold)<br />
1982 - Present DASSL Family<br />
1983 ODEPACK<br />
1985 Netlib goes online<br />
1986 Petzold, RK Methods for DAE’s<br />
1986 GMRES<br />
1988 Numerical Recipes Published<br />
1988 Mathematica 1.0<br />
1989 VODE<br />
1994-1996 CVODE/PVODE family<br />
1999 IDA (parallel version of CVODE/PVODE)<br />
2005 Sundials<br />
c○2007, B.E.<strong>Shapiro</strong><br />
Last revised: May 23, 2007<br />
Math 582B, Spring 2007<br />
California State University Northridge
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