Finite Difference Schemes

The basic idea of finite difference schemes to numerically approximate PDEs is to take your PDE, then transform the partial derivatives into finite differences of the exact solution by Taylor expanding, and then replace the Taylor expanded terms by the numerical approximations. I'm a little sloppy here, but we should also prove consistency (in the sense that as the finite differences \(\Delta x\to 0\), \(\Delta y\to 0\), etc., we recover the exact solution from the numerical approximation).

We should check for stability (possibly comparing results from implicit methods like backward Euler).

For instabilities arising from finite differences, see Rouson and friends "A grid-free abstraction of the Navier-Stokes equation" (2008) wherein are "provided a more subtle example inwhich information that appeared to have been lost during the discretization process was actually missing from the original PDE."

After reading the paper carefully, it seems Rousson and friends do the following:

1. Review a version of the incompressible Navier-Stokes equation. Specifically, one which reduces down to 2 equations in 2 unknowns.
2. Consider the Fourier spectral method for numerical analysis.
3. Show the equations from step 1 have problems at wavelength \(\vec{k}=0\), which reflects a loss of information.
4. Assert this loss of information occurs in every consistent scheme.

• DWI Rouson, R Rosenberg, X Xu, I Moulitsas, and SC Kassinos,
  "A grid-free abstraction of the Navier-Stokes equation in FORTRAN 95/2003."
  ACM Transactions on Mathematical Software 34, 1 (2008) 33 pages. doi:10.1145/1322436.1322438
• J.W. Thomas,
  Numerical Partial Differential Equations: Finite Difference Methods.
  Springer, 1995.