Adds new examples for simulating linear advection equation (David Ren)

[drdren]

Uploaded the tutorial Jupyter notebook with some images of the code's output, with comparison against the analytical solution.

• Please check if the PR fulfills these requirements

• The commit message follows our guidelines
• Tests for the changes have been added (for bug fixes / features)
• Docs have been added / updated (for bug fixes / features)

• What kind of change does this PR introduce? (Bug fix, feature, docs update, ...)
  Examples

• What is the current behavior? (You can also link to an open issue here)
  The open issue is: Program your favourite Hamiltonians in the OQD Analog and Atomic interfaces Program your favourite Hamiltonians in the OQD Analog and Atomic interfaces #60

• What is the new behavior (if this is a feature change)?
  This PR adds a tutorial for how to use OQD to solve the linear advection equation along with a couple of images of the results of the solution, with comparison against the analytical solution. The original paper's arxiv link is in the tutorial. The tutorial ends with a conclusion and some future directions. The tutorial itself is oqd_1D_v2.ipynb.

• Does this PR introduce a breaking change? (What changes might users need to make in their application due to this PR?)
  N/A

• Other information:
  Thank you very much! This was a highly enjoyable challenge. I appreciated the opportunity to learn about OQD's mission, tech, and philosophy. I would be interested in expanding this little tutorial further to cover other partial differential equations, or to write a package using OQD. Cheers!
  -David

[drdren]

Hi @benjimaclellan, thanks for this great opportunity to contribute to OQD as part of Unitary Hack 2025. I was interested in showing how OQD could be used to solve classical partial differential equations efficiently in an analog quantum computing framework. Please find the tutorial in oqd_1D_v2.ipynb, with some images of the quantum simulation's results with comparison against the analytical solution. I'd be very interested in extending this further to other partial differential equations. Cheers!
-David

[yhteoh]

@drdren Thanks for contributing to OQD, we would be interested in including additional examples for other PDEs.

[drdren]

Thanks, @yhteoh! I certainly will. How do you recommend that I submit new examples in the future?

[yhteoh]

Create a new PR on this repo (describe the example in the title) with the examples and tag me as a reviewer.

[drdren]

Thanks! I will do so! For reference, is there any (even tentative) date for an upcoming version update that I could use a deadline?