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% MATHEMATICS 507: SCHEDULE of CLASSWORK and ASSIGNMENTS %
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INSTRUCTOR: Roger Alexander 444 Carver (29)4-7579
CONSULTING HOURS: Monday, Tuesday 12:30-2:30; Thursday 3:00-4:00.
NOTE: Anything more than two weeks in the future is subject to change!
REQUIRED TEXT:
[AMR] U. Ascher, R. Mattheij & R. Russell, Numerical Solution of
Boundary Value Problems for Ordinary Differential Equations,
SIAM Classics, Philadelphia, 1995.
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%% OVERVIEW: %%
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%% I.METHODS FOR INITIAL VALUE PROBLEMS (7 weeks) %%
%% A. EULER'S METHOD (1 week) %%
%% B. ONE-STEP METHODS (1 1/2 weeks) %%
%% C. MULTISTEP METHODS (2 weeks) %%
%% D. EXTRAPOLATION METHODS (1/2 week) %%
%% E. STIFF PROBLEMS (2 weeks) %%
%% II. METHODS FOR BOUNDARY-VALUE PROBEMS (8 weeks) %%
%% A. BACKGROUND ON BOUNDARY-VALUE PROBLEMS (1 week) %%
%% B. INITIAL VALUE METHODS (4 weeks) %%
%% C. FINITE-DIFFERENCE METHODS (3 weeks) %%
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==week 1== EULER'S METHOD
27 AUG Euler's Method--a tool illustrating all the ideas
29 AUG Convergence of Euler's method
==week 2== ONE-STEP METHODS
3 SEP Introduction to Runge-Kutta methods
5 SEP Order conditions for Runge-Kutta
==week 3== =======
10 SEP Implementation of Runge-Kutta codes:
Automatic stepsize control (Richardson,Fehlberg)
Dense output
12 SEP Introduction to multistep methods:
Consistency, Stability,
First Problem Set due.
==week 4== MULTISTEP METHODS
17 SEP Convergence of multistep methods
First Dahlquist barrier
19 SEP Derivation of Adams & BDF formulae
Predictor-corrector algorithms
==week 5= ==========
24 SEP Adams methods in Nordsieck and Modified
Divided Difference forms.
Control of stepsize and order
26 SEP EXTRAPOLATION METHODS:
Extrapolation of Euler's method
==week 6== STIFF PROBLEMS
1 OCT The Gragg-Bulirsch-Stoer method
Second Problem Set due.
3 OCT Characterization of stiffness;
Singular-perturbation examples.
A-stability of numerical methods
==week 7== =======
8 OCT Second Dahlquist Barrier; BDF
10 OCT Runge-Kutta methods
==week 8== BOUNDARY-VALUE PROBLEMS
15 OCT Introduction to boundary-value problems [AMR] 3.1
17 OCT Green's functions, condition numbers [AMR] 3.2.1
First Computing Project Due
==week 9== INITIAL VALUE METHODS
22 OCT The simple shooting method: linear problems
24 OCT Instability of simple shooting
=week 10=
29 OCT The shooting method for nonlinear problems.
31 OCT Multiple shooting: introduction [AMR] 4.1, 4.2.1
=week 11= ==========
5 NOV Multiple shooting: theory [AMR] 4.3
7 NOV Multiple shooting: solving the linear system
=week 12=
12 NOV Multiple shooting for nonlinear problems [AMR] 4.6
14 NOV Nonlinear differential equations
Third Problem Set due.
=week 13= FINITE-DIFFERENCE METHODS
16 NOV Introduction to difference methods [K] Ch. 2
[AMR] 5.1
18 NOV One-step schemes for first order systems [AMR] 5.3
========== THANKSGIVING BREAK ============
=week 14=
30 NOV Collocation methods [AMR] 5.4
2 DEC Consistent + Stable <=> Convergent [AMR] 5.2
=week 15=
7 DEC Staircase matrices; solution of linear systems [AMR] 7.1-2
9 DEC Review
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