Main objectives

My main research objectives for the semester are:

1. Complete and submit paper on 2D cylindrical stegotons with Manuel Quezada
   • Similarity relations
   • Interactions
   • Writing (clean up final sections)
2. Complete and submit paper on many-stage integrators for spectral difference methods with Matteo Parsani
   • Address internal instability
   • Writing
3. PyClaw development:
   • Integration of code developed/started at the HPC^3 workshop, including DG, WENO
   • Refactor main classes to provide a streamlined user interface for the most common uses
   • Improve documentation
     • Better gallery pages
     • Better developer documentation
   • Improve testing
     • Continuous integration server setup (with Aron Ahmadia)
     • Use examples as tests
     • Greater code coverage
4. A general convergence proof for the closest point method.

I’m also providing support for the following projects:

• Development of AMR in PyClaw (PeanoClaw, p4est/PyClaw)
• SSP explicit multistep Runge-Kutta methods (with Sigal Gottlieb)
• Strong stability preserving effective order Runge-Kutta methods (Yiannis Hadjimichael)
• Implicit shallow water on the sphere in PyClaw (Matteo Parsani, Jed Brown, Kyle Mandli)

Other objectives

I will likely spend time revising and resubmitting the following papers:

• PyClaw (SISC)
• SharpClaw (SISC)
• Explicit RK stability optimization (SIREV)

Some ideas that I’m exploring which might become full-fledged projects later in the semester are:

• Development of explicit Runge-Kutta methods with many stages
  • Specified stability polynomials
  • Internally stable
  • Low-storage
• Accuracy optimization of Runge-Kutta methods
• Stability optimization of other classes of time integrators
  • Implicit RK
  • Stability optimization of linear multistep methods
• Urban flood modeling with PyClaw. Requirements:
  • Topography of Jeddah and surroundings
  • GeoClaw Riemann solvers in PyClaw (Kyle Mandli)
• Multi-time integrators (w/Colin Macdonald, Steve Ruuth)

• Implicit SSP methods, including multistep multistage, high order downwind, and some good example problems (with Sigal Gottlieb)

• Methods for seismic imaging (with Tariq al-Khalifah)

• Chaos in quasi-periodic waves in periodic media

Week-by-week outline

• February 18-29:
  • Work with Zoltan Horvath on positivity preservation and the radius of monotonicity
  • Extract and analyze shape information from cylindrical stegoton data
  • Writing for SD and stegoton papers
  • Final review of Amal’s thesis
• March 3-7: Work on Entropy dissipation schemes with Bojan Popov
• March 10-14: Prepare lectures and projects for FIC spring school
  • SSP Theory
  • NodePy
  • PyClaw
  • Positivity preservation for shallow water or Euler
• March 17-25: Run FIC Spring school on positivity
• March 26 - April 4: PyClaw development
  • AMR
  • Refactoring
  • Testing
• April 7-11: Work on stegoton stability with Jeff Humpherys
• April 16-25: Travel to Texas A&M (for work on hyperbolic solvers), CWI (for work on positivity preservation)
• April 28 - May 2 Vacation/travel
• May 5-9, 12-16: Accuracy optimization of RK methods
  • Test simple examples
  • Draft report

Review and accomplishments

Completed:

1. PyClaw paper resubmitted and accepted to SISC
2. Major refactoring of PyClaw (UI, testing, Peano interface)
3. FIC Spring school (paper in preparation)

Progress made:

1. Cylindrical stegotons paper - new approach to interactions

• Goal: submit in June

2. Effective order SSP - lots of revising

• Goal: submit as soon as Jim Verner reviews

3. Optimized RK for SD - writing and improving RK-opt code

• Goal: submit as soon as numerical tests are complete

4. Several beginning projects:

• Combinatorial approach to waves in layered media (with Bojan Popov)
• Multi-time integrators (with Macdonald, Ruuth)
• Sketch of proof of global optimality of stability polynomial optimization
• Literature review of semi-infinite programming