NUMERICAL SOLUTION PARTIAL DE (MTH_453_X010_S2024)
== week 1

1. 4/1/24: Introduction. See Syllabus. PDE examples: classification and overview.

See Class notes MTH_453_553_notes.pdf. See also the page Resources. 

2. 4/3/24: Why it is best to not use chain rule, product rule, and such when approximating solutions to a PDE.

Truncation error vs round-off error when approximating  derivatives.

Back to BVP in 1d: Neumann condition. 

3. 4/5/24: More on BVP Neumann probles: comparison of error analysis, stability, well-posedness. 
Q1 due.

== week 2

4. 4/8/24: Overview of the PDEs: transport equation, heat/diffusion equation, Laplace/Poisson equation, and wave equation. Elliptic/parabolic/hyperbolic PDEs. Conservative and dissipative PDEs. Numerical examples how the solution changes from initial condition. Ideas how the solutions behave on simple 1d examples. 

Extra office hours 4:00-5:30

5. 4/10/24: Solving BPV in 1d generalized to solving Poisson's equation in 2d. Outline: accuracy (LTE) and stability follow analogously as in 1d.  

6. 4/12/24: Solving Poisson's equation on unit square with (FD): details of organization of the algorithm. Numbering of grid unknowns. 

HW1 due. 

== week 3

7. 4/15/24: Details of implementation of FD for Poisson's equation. More general elliptic PDEs. 

Extra office hours 4:00-5:30

8. 4/17/24: NO IN CLASS MEETING: watch FD for parabolic PDE: theory

Most of the theory and examples are covered in [TEXTBOOK, Chapter 9].

Office hours cancelled: please contact the instructor by email, to set up zoom meeting of needed. 

9. 4/19/24: FD for parabolic PDEs: theory and algorithm.


HW2 due 

== week 4

10. 4/22/24: FD for parabolic PDE: implementation

11. 4/24/24: continue FD for parabolic problems: LTE. 

12. 4/26/24: stability analysis via MOL.

Q2 due.

== week 5

13. 4/29/24: Wrap-up stability analysis with method of lines for heat equation and reaction-diffusion equation. Recall eigenvalue/eiegnfunction/eigenvalue analysis for diffusion operator under Dirichlet conditions. 

14. 5/1/24: Convergence proof for FD discretization of heat equation combining stability and consistency: worksheet for Forward Euler (FE). 

EXTRA (MAKE_UP) CLASS scheduled in STAG 160, 4:00-5:30

Worksheet for BE; discussion of CN scheme. Lax-Richtmyer stability + consistency give convergence.

Introduce von-Neumann Ansatz: example for calculation of growth factor and stability analysis  for FE scheme. 

15. 5/3/24: review. 

== week 6

16. 5/6/24: MIDTEM in class. 

EXTRA office hours 4:00-5:30.

17. 5/8/24: How to solve problems with inhomogeneous coefficients. How the heat equation is derived (Fourier, Fick's, Darcy's laws). 

Why nodal FD is not good for heterogeneous problems in 2D.

18. 5/10/24: Alternatives to fully implicit schemes in 2D: locally 1D and ADI methods. 

Why exp(lambda x) is a useful eigenfunction of d/dx.  Fourier series: why useful in analysis of (partial) differential equations of second order. (Also, consider -u''=f with Dirichlet and periodic b.cond.)

== week 7

19. 5/13/24: More on Fourier analysis and von-Neumann Ansatz. Advection equation: introduction. (NO VIDEO RECORDING). 

EXTRA (MAKE_UP) CLASS scheduled in STAG 160, 4:00-5:30

Solving an advection equation: notion of auxilliary conditions, method of characteristics, and weak/distributional solutions. [CHAPTER 10]  Upwind and central scheme.

Group work on Von-Neumann stability analysis for these. 

HW 3 due. 

20. 5/15/24:  Generalization of u_t+au_x=0 to systems of conservation laws. The LTE and modified equation analysis for the (explicit) upwind scheme. 

21. 5/17/24:  Implicit upwind scheme: why and why not (stability, LTE, and modified equation). 

HW4 (midterm corrections) due. 

== week 8

22. 5/20/24: Finish stability analysis. Synthesis of schemes so far for the advection equation.  

23. 5/22/24: Lax-Friedrichs scheme (LF). How to handle the boundary conditions for advection. Recommended way to implement the schemes. 

24. 5/24/24:  More on notation and Dirichlet vs periodic b.c. MOL approach: why hard for Dirichlet b.c. and why von Neumann stability analysis does not work. 

== week 9

25. 5/27/24: MEMORIAL DAY (NO CLASS).

HW5 due 

26. 5/29/24: Error analysis for LF scheme for advection equation with periodic b.c.: perspectives gained from error analysis for BE/FE time discrete schemes for diffusion equation.  What is easy and what is not using L^2 norm. Error analysis for LF scheme in L^1 norm. 

27. 5/31/24: Worksheet on ADR (advection-reaction-diffusion) schemes. Demo of plotting stability for a chosen dt. 

== week 10

28. 6/3/24: Mixed equations [Chapter 11] approximated with operator splitting and IMEX schemes. Computational cost vs accuracy. 

29. 6/5/24: More on outflow vs inflow conditions in splitting schemes.  Schemes for second order wave equation, treated as second order PDE, and as a system of first order PDEs. 

What problems to practice with? 

30. 6/7/24: Wrap-up/review. 

HW6 due. 

== week 11  FINALS 

Tuesday 6/11/24 6:00-8:00pm.