Errors in Simple Handwritten PDEs as ODEs

[GodotMisogi]

The handwritten PDEs as ODEs benchmarks have been failing, particularly the pseudospectral ones (discussed in #929).

The issues to be addressed are:

1. Compatibility with the current Julia and package versions (v1.11, v0.8.0)
2. Resolve the implementation errors in the finite-difference and pseudospectral methods.
3. Determine appropriate ODE and linear solver choices for the benchmarks of both methods.
4. Provide documentation of the equations and the respective discretizations.