implement a simple C++ backward Euler integrator

[zingale]

[zingale]

this can integrate aprox13

it only uses rtol_spec and atol_spec for error estimation.

[zingale]

Here's a comparison using burn_cell between VODE and Backward Euler:

[maximumcats]

Can we use the linpack routines from util/?

[zingale]

yes, I just need to make that not depend on VODE. I'll do that in a separate PR.

[zingale]

okay, this builds and seems to work once again.

[zingale]

I was able to integrate both a pynucastro net and an aprox net with this integrator

[maximumcats]

Results of this integrator on the test in #632:

$ ./main3d.gnu.ex inputs_aprox13_mott unit_test.tmax = 1.e2
AMReX (22.11-14-ge0afc82f63ce) initialized
starting the single zone burn...
Maximum Time (s): 100
State Density (g/cm^3): 10000000
State Temperature (K): 5000000000
Mass Fraction (He4): 0
Mass Fraction (C12): 0.5
Mass Fraction (O16): 0.5
Mass Fraction (Ne20): 0
Mass Fraction (Mg24): 0
Mass Fraction (Si28): 0
Mass Fraction (S32): 0
Mass Fraction (Ar36): 0
Mass Fraction (Ca40): 0
Mass Fraction (Ti44): 0
Mass Fraction (Cr48): 0
Mass Fraction (Fe52): 0
Mass Fraction (Ni56): 0
RHS at t = 0
   He4 18982072.69
   C12 -37920721.9
   O16 -483268.4495
  Ne20 18722989.58
  Mg24 237221.6847
  Si28 239033.521
   S32 206.7673375
  Ar36 -6.80347759e-25
  Ca40 2.671937993e-24
  Ti44 -2.870067494e-24
  Cr48 -1.64899539e-26
  Fe52 -7.334826867e-27
  Ni56 -3.889639522e-27
integration summary:
dens: 10000000 temp: 5000000000
energy released: 6.541799691e+17
number of steps taken: 6074
number of f evaluations: 26408
------------------------------------
successful? 1
 - Hnuc = 6.541799691e+15
 - added e = 6.541799691e+17
 - final T = 5000000000
------------------------------------
e initial = 1.328677169e+18
e final =   1.982857138e+18
------------------------------------
new mass fractions:
He4 0.04129594208
C12 1.891873058e-07
O16 7.519834653e-07
Ne20 9.083521524e-09
Mg24 4.072337551e-06
Si28 0.008131179773
S32 0.01341589955
Ar36 0.01043788741
Ca40 0.01983642858
Ti44 0.000434485126
Cr48 0.003588685605
Fe52 0.05204526759
Ni56 0.8508092017
------------------------------------
species creation rates:
omegadot(He4): 0.0004129594208
omegadot(C12): -0.004999998108
omegadot(O16): -0.00499999248
omegadot(Ne20): 9.083521524e-11
omegadot(Mg24): 4.072337551e-08
omegadot(Si28): 8.131179773e-05
omegadot(S32): 0.0001341589955
omegadot(Ar36): 0.0001043788741
omegadot(Ca40): 0.0001983642858
omegadot(Ti44): 4.34485126e-06
omegadot(Cr48): 3.588685605e-05
omegadot(Fe52): 0.0005204526759
omegadot(Ni56): 0.008508092017
number of steps taken: 1070229712
AMReX (22.11-14-ge0afc82f63ce) finalized

This matches pretty closely. It's an order of magnitude more timesteps than VODE, but several orders of magnitude fewer timesteps than the forward Euler integrator.