Add tridiagonal solver design document

[mwarusz]

Design document for tridiagonal solve, which will be used for the vertical diffusion in the momentum and tracer equations.

Checklist

• Documentation:
  • Design document has been generated and added to the docs
  • Documentation has been built locally and changes look as expected

[cbegeman]

Note that this design doc includes the Schopf and Loughe (1995) approach for accuracy and stability discussed here: E3SM-Ocean-Discussion#42

[mwarusz]

@mark-petersen

I have updated this design doc and it is now ready for review. I also opened a draft PR with the implementation (#194). When reviewing the design it might be helpful for you to have a look at one of the unit tests showing how these solvers are used to solve a simple 1D diffusion problem:

https://github.com/mwarusz/E3SM/blob/5617516be139facfaa904e0713287a0d9e4d5b8f/components/omega/test/base/TridiagonalTest.cpp#L146-L245

[mark-petersen]

@mwarusz, thank you for this detailed design document. I compiled locally and reviewed the html version, which looks good. Thanks for adding the Thomas and parallel cyclic reduction versions, which will be useful for stability and performance.

This design doc begins with the equation for the general tridiagonal solve, which is the right scope for this part of the project. In a future design doc we can specify the form of those three coefficients and the boundary equations, as I did here for MPAS-Ocean.

[mark-petersen]

Pinging @katsmith133 @vanroekel and @philipwjones, if any of you want to review this design document.

[philipwjones]

Seems ok to me. I may be more particular about the interface above this for mixing/diffusion.

[vanroekel]

Also looks fine to me.