Math 269A


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Course: Math 269A, Advanced Numerical Analysis

Text: A. Iserles, A first course in the Numerical Analysis of Differential Equations", Cambridge Texts in Applied Mathematics, Cambridge University Press, 1998

Schedule of Lectures:

Notations and terminology for ODE's and systems of ODE's; reduction of higher order ODE's to 1st order systems of ODE's; the fundamental existence and uniqueness thm. for ODE's (Lipschitz condition)

Introduction of Euler's method, order of Euler's method, one step methods (introduction, definition, consistency, local truncation error)

Explicit Runge-Kutta (ERK) methods (introduction of the method in the general case, notations in the general case, derivation of ERK of second order); Runge-Kutta method of fourth order

Examples of implicit methods: trapezoidal rule, midpoint rule, the theta method, and the implicit Euler's method; computation of orders for these methods

Convergence of one-step methods (the general case; see also convergence for Euler's method, etc)

Asymptotic expansions for the global discretization error for one step methods, and applications to error estimate

Practical implementation of one step methods

Linear Multistep methods: examples, derivation using the Lagrange interpolation polynomial

Linear multistep methods: definition and computations of the local truncation error, order of the method, consistency

Implicit and explicit linear multistep methods, predictor-corrector methods

Examples of consistent multistep methods which diverge

Linear difference equations: stability (root) condition, general solution

Convergence Thm. for linear multistep methods

Order and consistency for linear multistep methods

Adaptive methods for one-step and multi-step methods, error control, Milne device, extrapolation

Stiff differential equations, stability and intervals (regions) of absolute stability, A-stable methods, BDF methods

Numerical methods and stability for systems of ODE's

Finite difference methods for linear BVP

Functional (fixed point) iteration and Newton's iteration for solving systems of ODE's using an implicit method