NUMERICAL SOLUTION ORDINARY DE (MTH_452_X010_W2024)
== week 1

1. 1.8/24: Introduction. See MTH_452-552-Syllabus.pdf. First demonstrations and examples. See Class Notes in  MTH_452_552_notes.pdf 

See also the page RESOURCES.

2. 1/10/24: Examples. First algorithms (FE and BE), Lipschitz condition.  [TEXTBOOK Chapter 5]

Wednesday 1/10/24 at 4pm: MATLAB clinic in MSLC computer lab (Kidd 108).

3. 1/12/24: Well posedness, stability, Gronwall's lemma.  Implementation of schemes, and error calculations. 

DUE: Q00 and Q01. 

== week 2

*** 1/15/24: No class (MLK holiday)

DUE Wednesday 1/17/24: HW1. See  General guidelines on Homework. 

4. 1/17/24: No class (Campus closed)

Wednesday 1/17/24 at 4pm: Second MATLAB clinic on zoom.

HW1/452 due. also HW1/552 due.

5. 1/19/24: Local truncation error (LTE). Tools: Taylor expansions. Method is consistent if LTE=O(h^r), r>=1. Examples: LTE of FE, BE is O(h). Trapezoidal (implicit and explicit=Heun), and midpoint (implicit and explicit) methods are O(h^2).

== week 3

6. 1/22/24:  Examples how to calculate/estimate LTE. Handouts with a tableau. 

Monday at 5pm: Third MATLAB clinic in MSLC computer lab (Kidd 108).

7. 1/24/24: Another look at accuracy when f(u)=\lambda u (compare scheme to exponential). Start multi-stage methods (Butcher tableaus for Runge-Kutta schemes). 

Wednesday at 2pm: Fourth MATLAB clinic in MSLC computer lab (Kidd 108).

8. 1/26/24: Examples of RK schemes and checking consistency. 

HW2 due. (Corrections to HW1 due as extra credit). 

== week 4

9. 1/29/24: Examples of RK: constructing and deconstructing. Explicit and implicit and diagonally implicit schemes. Checking consistency of RK4. LTE when f=f(t,u). (Must use bivariate Taylor expansion). 

10. 1/31/24: How to define higher order schemes. Richardson extrapolation, and Taylor series. Multi-step schemes. Example: AB-2.  

11. 2/2/24: Multi-step schemes. How to derive Adams family methods based on approximating f(t,u(t)) with a polynomial. General LMM framework. Consistency of LMM, and examples. 

HW3 due. 

 See HW1-3 notes video. 

== week 5

12. 2/5/24: LMM with characteristic polynomials.  Error estimate for FE scheme, u'=lambda u. [TEXTBOOK Chapter 6]

Convergence <==> Consistency & Zero-stability.

13. 2/7/24:  Zero-stability: root condition for characteristic polynomial. Why zero-stability is necessary for convergence.

14. 2/9/24: REVIEW for MIDTERM. 

Extra office hours Friday 1:00-3:00pm. 

== week 6

15. 2/12/24: MIDTERM (in class). 

16. 2/14/24: Lecture on solving linear ODE systems is here Lecture 2/14/24 on linear systems

Query mid-semester is due. 

17. 2/16/24: Lecture on solving implicit schemes is here Lecture 2/16/24 on solving implicit schemes

== week 7 MATLAB clinic in MSLC computer lab (Kidd 108) will be available, schedule Monday at 4pm, Wednesday at 1pm.

18. 2/19/24: Error calculations for FE, nonlinear problem, and for the general single-step scheme. Different ways the error accumulates (sharp and less sharp estimates). 

But, error is not everything! [TEXTBOOK Chapter 7] How some schemes (FE) do not honor the qualitative properties of the true solution. Need additional stability criteria.  

19. 2/21/24: Absolute stability: for test equation, and lambda <=0, decay of numerical solutions for BE, and only conditional decay for FE. 

Finding growth factors for single-step schemes: Heun and MIDTERM scheme. 

20. 2/23/24: Absolute stability for LMM. 

HW4 due  and extra credit for HW4 .

== week 8

21. 2/26/24: How to plot absolute stability regions. 

22. 2/28/24: A-stability and L-stability. How to choose a time step (when given a scheme): competition of accuracy and efficiency. How to estimate computaitonal cost of a scheme. 

23. 3/1/24: How to estimate LTE a-posteriori (embedded methods [TEXTBOOK Section 5.7.1]: e.g, FE and Heun). How to estimate global error a-posteriori (Richardson extrapolation). 

==  week 9

24. 3/4/24: Boundary value problems of second order ODEs. TEXTBOOK Sections 2.1-2.15] Basic scheme for u''=f. LTE for the second order difference operator requires C^4 smoothness. 

25. 3/6/24: Solving Dirichlet BVP: linear system with a symmetric positive matrix. Examples of finding true solution, e.g., with a piecewise function f. 

26. 3/8/24: Accuracy and stability analysis for BVP problems. (Using vector and matrix norms).

(Optional) material on the eigenvalues of differential operator in BVP. 

HW5 due. 

== week 10

27. 3/11/24:  Review. 

Extra office hour 11:00-11:50 and 4:00-4:50pm.

28. 3/13/24: No in-person class.  (Office hours cancelled)

Lectures week 10

(You will find there the additional material on vector/matrix/function norms, and the applications of ODEs, as well as solving systems of ODEs, the understanding of stiff systems, and the final connection between the BVP and IVP inan IBVP for a heat equation). 

29. 3/15/24: No in-person class. (Office hours cancelled)

HW6 due 

===================

Monday  3/18/24 OFFICE HOURS. 3:00pm-5:00pm in my office Kidd 298B or TBA. 

Tuesday 3/19/24: Final Exam.