david/Spring 2023 plans

Projects to focus on this semester

Manuscripts to finalize and submit

Manuscripts I expect to revise & resubmit:

Teaching

Research projects

These are far enough along that I hope to have a paper submitted by the end of the semester:

These are more preliminary, but I hope to have a draft of a manuscript by the end of the semester:

These are projects where I’m in an advisory role:

These are just ideas at the moment, but I hope to find time to start exploring them:

  • Long-time behavior of SW & Euler solutions in 1D (globally regular solutions?)
  • Homogenization of the Euler equations

Week-by-week plan

  • Jan. 29-Feb. 2:
    • [X] Finish revising and resubmit WSO structure manuscript;
    • [X] finalize and submit multiple relaxation KdV manuscript
    • [X] finalize and resubmit iterative Riemann solvers manuscript
  • Feb. 4-12:
    • Work with Lajos Loczi on automating perturbation theory in Mathematica
    • Try to work out next-order terms for 1D and 2D shallow water homogenization
  • Feb. 12-17: Work on MPRK order conditions with Meister and Izgin
  • Feb. 17-26: ANODE meeting in Auckland, NZ
  • March:
    • Research and write MPRK manuscript – Thomas Izgin has taken the lead on this
    • Run additional experiments for 1D Shallow water homogenization
    • Complete 1D SW homogenization manuscript
  • March 12-16:
    • [X] Figure out when certain average functionals vanish, depending on the nature of the function
    • [X] Re-derive 1D SW homogenization in a more automated way in Mathematica (done up to \(O(\delta^2)\))
  • March 19-23:
    • [ ] Re-derive 2D SW homogenization in a more automated way
    • [ ] Incorporate existing material from collaborators’ notes into WSO ERK manuscript
  • March 26-30:
    • [ ] Conduct detailed numerical comparisons of 1D SW homogenized equations
    • [ ] Investigate oscillation that arise behind the wave train
    • [ ] Try solving both xxx and xxt forms
  • April 2-6:
    • Flesh out 1D SW homogenization manuscript
  • April 9-13:
    • Run additional experiments for 2D shallow water homogenization

  • April 16-18:

  • April 19-29: Eid

  • May:
    • Finalize and submit SW homogenization manuscripts
    • Work on MPRK manuscripts
    • Maybe finalize and submit ERK WSO manuscript

  • Semester ends May 14

End of semester review

I’m happy with the material developed for the nonlinear dispersive waves course. We didn’t cover everything I hoped to, but I learned a lot.

Status of my own research projects:

  • Shallow water homogenization papers: together with Lajos Loczi, I developed a almost-fully-automated process, and re-derived the 1D equations with it. The re-derivation of the 2D equations is work in progress. I didn’t get to numerical simulations or further writing. I may put these on hold until when Giovanni Russo is at KAUST in the fall.
  • MPRK paper (with Meister and Izgin): this became two papers; the first is 95% finished and will probably be submitted next week. After that we will start the second.
  • Explicit RK weak stage order: I’m still in the process of organizing and rewriting all the material developed by my collaborators for this paper. Hope to finish in the next couple of weeks and submit the paper in June/July.
  • Hyperbolization of high-order PDEs: this is a new project that grew out of my teaching for AMCS 394D. The paper is in a rough draft state and I hope to complete it in the next few weeks.