Lecture 22, 18.300, Tue Apr 26, 2022.
Topic: numerical solution of evolution pde. Computer lecture.
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Summary:
1) Formulation of a simple FD numerical scheme for the wave equation in 1-D.
2) Computer illustration of what the simple scheme does. This was done with
   the GBNS_lecture.m in the course Toolkit.   Note: this script uses a GUI
   that may not work with your version of MatLab. If so, run lectureGBNS.m,
   which uses no GUI. Modulo minor fixes, these script should work with any
   version of MatLab.
3) The simple scheme is unstable. A fix using ``artificial viscosity'' is
   used to produce a convergent scheme [the ``good'' scheme in the scripts].
   The idea/reason for this fix is explained in [R4] and [R5] below.
4) Theory. Conditions that a scheme must satisfy to be useful:
   Consistent \ These is explained in [R2] and [R3] below.
   Stable     /
   These are necessary for convergence [see (5)].
   For practical use: efficiency is also a must. Convergence must happen as
   fast as possible.
5) Lax theorem: for linear pde/schemes, Consistency plus Stability gives
   convergence.
6) CFL (Courant-Friedrichs-Lewy) condition: a necessary condition for
   stability is: The numerical scheme domain of dependence must
                 include the actual domain of dependence.
   It should be clear why this is needed.
   This condition is not sufficient.
7) von Neumann Stability analysis: works for constant coefficients, linear,
   evolution pde.
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8) Discuss the Wiley E. Coyote problem.
   The issue of a finite speed of propagation matters, in many contexts:
   a) In seismology to identify the source of an earthquake by looking at
      the signal arrival time at various stations [triangulation]. This is
      wave propagation through a solid --- the Wiley E. Coyote story.
   b) The tidal wave produced by an earthquake at sea takes time to arrive
      to the coast. Predicting when and where matters, a lot.
   c) A radar can tell how far an object is by the time delay between
      sending and getting the reflected signal.
   d) You can tell how far lightning happens by measuring the delay from
      the light to the sound signal.
   e) Astronomers look at the past of the Universe by looking at far away
      galaxies, etc.
   f) Packing as tightly as possible the circuits in a computer increases
      computational speed. This is one reason why small is better there.
   g) You cannot have a robot-rover in, say, Mars directed by tele-presence
      because of the time delay.
   Think of others.
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Relevant notes in the Course Web page % <===================================
         [https://math.mit.edu/classes/18.300/Notes/index.html]
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[R1] Numerical solution of pde --- quick preview.
        Short summary of what is seen when running the scripts
        GBNS_lecture.m or lectureGBNS.m in the course Toolkit.
[R2] Stability of Numerical Schemes for PDE's.
        A detailed version of [R1], including a von Neumann stability
        analysis. Section 3 here is missing, check instead [R4-R5]
[R3] Various lecture notes for 18311 ...
     Convergence of Numerical Schemes ...
        See section 1 here for a more precise treatment of the topics.
[R4] Notes: von Neumann Stability Analysis.
[R5] Notes: Associated Equation to a Numerical Scheme.
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