System Transformation - Automatically make PDESystems compatible

[xtalax]

Allows

• Mixed derivatives
• Composite derivative arguments
• Time agnostic BC/IC parsing
• More free choice of ic equation structure
• Arbitrary order time derivatives
• Expandable derivative terms in equations
• Use ODAEProblem where appropriate

pde_system_tranformation.jl is where the meat of this happens

[xtalax]

@ChrisRackauckas Ok, this is actually "working" in that it's correctly transforming the system and adding the right new equations, but it's making solve time oppressively long.

I'm testing on periodic Dt(u) ~ Dxy(u), but this is unstable even with WENO and sensibly short timesteps. Can you suggest a better system to test?

[codecov]

Codecov Report

Merging #168 (cd7d172) into master (81de83c) will increase coverage by 24.64%.
The diff coverage is 83.61%.

@@             Coverage Diff             @@
##           master     #168       +/-   ##
===========================================
+ Coverage   61.89%   86.53%   +24.64%     
===========================================
  Files          31       33        +2     
  Lines        1412     1664      +252     
===========================================
+ Hits          874     1440      +566     
+ Misses        538      224      -314     

Impacted Files	Coverage Δ
...discretization/generate_finite_difference_rules.jl	90.90% <0.00%> (+18.18%)	⬆️
...hemes/centered_difference/centered_diff_weights.jl	100.00% <ø> (+50.00%)	⬆️
src/discretization/schemes/half_offset_weights.jl	100.00% <ø> (+48.64%)	⬆️
...n/schemes/upwind_difference/upwind_diff_weights.jl	97.29% <ø> (+48.64%)	⬆️
src/MOL_utils.jl	72.88% <40.00%> (-5.14%)	⬇️
src/discretization/generate_bc_eqs.jl	82.96% <50.00%> (+28.88%)	⬆️
src/interface/solution/timedep.jl	85.29% <71.42%> (+11.96%)	⬆️
src/system_parsing/variable_map.jl	83.87% <83.87%> (ø)
src/discretization/discretize_vars.jl	86.90% <84.37%> (+9.12%)	⬆️
src/MOL_discretization.jl	74.45% <86.79%> (+12.73%)	⬆️
... and 33 more

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[xtalax]

CI failures work locally, merging to get on with other stuff