Advanced Numerical Analysis
Math 269A,  Fall 98,  UCLA

Lecture:    MS 5138  MW 4-5:15pm

Instructor: Professor Eitan Tadmor  Office hours --  MS 7945, Tu:  2-3pm,  Fr: 2:30-3:30pm

Dicussion:  MS 5138  F 1-1:50pm

Teaching assistant: S. Nezzar           Office hours -- MS 6147,  We: 12:30-1:30, Tr: 10-10:45

Content:

    Introduction: systems of ODEs. Existence, uniqueness & stability.

    Notes #1: Stability bounds for nonlinear systems of ODEs [ HTML file ] [ pdf file ]

    Euler's method.

    Notes #2: Convergence rate estimates for Euler scheme [ HTML file ] [ pdf file ]

    Multistep methods. The example of Milne schemes.

    Assignment #1Recursion schemes and error bounds [ HTML file ] [ pdf file ]

    Predictor -- Corrector methods. Adams-Bashforth-Moulton schemes.

    Notes #3: Adams-Bashforth-Moulton Schemes [ HTML file ] [ pdf file ]

    Assignment #2 ODEs solvers using predictor-corrector Adams schemes [ HTML file ] [ pdf file ]

    Consistency, stability and convergence. Dahlquist theory. The root condition.

    Notes #4: Matrices: Eigenvalues, Norms and Bounded Powers [ HTML file ] [ pdf file ]

    Assignment #3 Order of accuracy and the characteritic polynomail [ HTML file ] [ pdf file ]

    Convergence of multi-step schemes

    Notes #5: Convergence of Multi-Step Schemes [ HTML file ] [ pdf file ]

    Local time stepping and error estimates.
    Runge-Kutta methods. Examples of RK4 and RKF5 schemes.

    Notes #6: Stability and Convergence of Runge-Kutta Schemes [ HTML file ] [ pdf file ]

    Assignment #4 Runge-Kutta schemes [ HTML file ] [ pdf file ]

    Notions of stability and stiff equations.


Final (Few answers included) [ HTML file ] [ pdf file ]

References:


E. Hairer, S.P. Norsett and G. Wanner, Solving ODEs I: Nonstiff Problems (1991, 2nd ed)., Springer-Verlag, Berlin.Everything - the modern version.
M.K. Jain, Numerical solution of DEs (2nd ed.), Halsted Press.
A. Iserles, A first course in the numerical analysis of DEs, Cambridge Texts.
Gear, Numerical initial value problems in ODE's (1971). The classical reference on theory and applications.
Lambert, Computational methods for ODE's (1991). Detailed discussion of ideas and practical implementation.
Shampine and Gordon, Computer solution of ODE's (1975). Adams methods and practial implementation of ODE "black box" solvers.
Butcher, Numerical analysis of ODE's (1987). Comprehensive discussion on Runge-Kutta methods.