NumericalMethodNotes.txt

In Galerkin finite elements, the mass matrix is not diagonal. Mass lumping is a procedure in which all
the elements in a mass matrix row are summed, and then that sum is put on the diagonal. All off-diagonal
elements are then set to zero. 

The explicit Forward Euler method for finite element is not stable unless I use mass lumping. This is mysterious to me. 
Obtained good result for Forward Euler with Mass Lumping with dt = h**2/2 (Both with old code and with new code that combines Forward Euler to Backward Euler much more neatly)
Obtained good result for Backward Euler with Mass Lumping with dt = h
Obtained undamped oscillatons for Forward Euler with Mass Lumping with dt = h (Here we see the explicit time stepping limitation)
Obtained damped oscillations for Crank with Mass Lumping with dt = h
Obtained good result for Crank with Mass Lumping with dt = h**2/2
Obtained good result for Backward Euler without Mass Lumping with dt = h**2/2
Obtained good result for Backward Euler without Mass Lumping with dt = h
Obtained damped oscillations for Crank without Mass Lumping with dt = h
Obtained good result for Crank without Mass Lumping with dt = h**2/2
Obtained good result for Forward Euler without Mass Lumping with dt = h**3/2 (This eliminates the idea that Forward Euler simply won't work without mass lumping. It's just that the stability criterion for dt is very strict for this numerical scheme. Good.) 

Conclusions:
Forward Euler only good with Mass Lumping and Small Time Steps (mass lumping essential)
Backward Euler good regardless of time stepping or mass lumping (mass lumping appeared to have no effect)
Crank good for small time steps, damped oscillatory for large time steps (mass lumping appeared to have no effect)
Good behavior seems to be independent of mesh size

Simpson's rule is exact for estimating the integral of second order polynomials. The trapezoidal rule would be exact for estimating the integral of first order polynomials.

My brief experiments wuld lead me to conclude that Gaussian quadrature can exactly approximate integrals of polynomials as long as the number of points >= the polynomial order. This kind of seems like it would make intuitive sense.

With 0.7 for the cfl_factor, sim fails at 1.61 ns. With 0.1 for CFL factor, sim fails at 2.34 ns. This was actually a quicker fail than when I didn't have downwinding for the electron_energy.