Keaton J. Burns

Hello! I'm an applied math instructor at MIT working on scientific computing and fluid dynamics. I'm developing the spectral PDE solver Dedalus and utilizing it to study diverse problems in astrophysical, geophysical, and biological fluids.

Research


Dedalus PDE solver

I'm the lead developer of Dedalus, an open-source framework for solving partial differential equations with modern spectral methods. Dedalus uses symbolic model specification to produce spectrally accurate, optimally sparse, and automatically parallelized solvers for broad ranges of custom equation sets on simple domains. It's written in Python and is easy to use on a laptop, yet calls compiled libraries for performance-critical routines and scales to thousands of cores with MPI.

Collaborators: Ben Brown, Daniel Lecoanet, Jeff Oishi, Geoff Vasil
References: Burns et al. (2020)


Modern spectral methods for PDEs

As a part of the Dedalus project, I am developing new algorithims applying sparse spectral methods to large-scale scientific PDEs. My work in this area includes:

  • Fast direct solvers for coupled differential-algebraic systems.
  • Generalized tau methods for enforcing boundary conditions.
  • High-order immersed boundary and domain remapping techniques.
  • Coupling to other PDE and BIE solvers.

Collaborators: Dedalus team, Dan Fortunato, Eric Hester, Manas Rachh


Steller convection and the solar dynamo

I am a co-I on a NASA-funded project studying solar activity and dynamo generation using Dedalus. We are performing local and global simulations to study the magnetohydrodynamic stability and evolution of the solar near-surface shear layer. Funding is available for MIT students interested in PDE-based optimization for this project.

Collaborators: Dedalus team, Kyle Augustson


Nonlinear tides in neutron stars

As a part of my thesis work, I used Dedalus to study nonlinear tidal effects in astrophysical binaries. I used direct numerical simulations to test theories of nonlinear tidal instabilities and dissipation in inspiralling neutron star binaries, like those recently observed by LIGO and VIRGO.

Collaborators: Nevin Weinberg
References: Burns (2018)


Glacial meltwater plumes

As a part of my thesis work, I used Dedalus to model turbulent meltwater plumes at the edges of marine-terminating glaciers. I examined how the thermal and compositional structure of the ambient water influences the turbulent heat flux in these plumes, which may influence the melt rate and evolution of the glacier.

Collaborators: Glenn Flierl, Andrew Wells
References: Burns (2018)


Rolling resistance on granular media

During the 2016 Geophysical Fluid Dynamics program at the Woods Hole Oceanographic Institute, I performed experiments examining the dynamics of rolling objects on sand and other granular materials.

Collaborators: Neil Balmforth, Ian Hewitt
References: Burns et al. (2017)

Recent Teaching

Fall 2022
MIT 18.336J/6.7340J: Fast Methods for Partial Differential and Integral Equations

18.336 is a graduate course, cross-listed between math and computer science, covering preconditioned finite difference methods, Fourier and polynomial spectral methods, and low-rank methods for PDEs and integral equations. See the course page on github: https://github.com/mitmath/18336.

IAP 2022
MIT 18.031: System Functions and the Laplace Transform

18.031 is an undergraduate course covering continuous control theory and representations of functions in the complex frequency domain. Includes generalized functions, unit impulse responses, convolutions, Laplace transforms, system/transfer functions, and pole diagrams. Includes examples from mechanical and electrical engineering.

Publications

Journal publications & preprints

  • "Emergent universal statistics in nonequilibrium systems with dynamical scale selection"
    Heinonen et al., submitted to Nature Physics, 2022. [arxiv]
  • "Effective drag in rotating, poorly conducting plasma turbulence"
    Benavides et al., accepted to Astrophysical Journal, 2022. [arxiv]
  • "Direct Statistical Simulation of the Busse Annulus"
    Oishi et al., accepted to Journal of Fluid Mechanics, 2022. [arxiv]
  • "Transport and emergent stratification in the equilibrated Eady model: the vortex gas scaling regime"
    Gallet et al., accepted to Journal of Fluid Mechanics, 2022.
  • "Inverse cascade suppression and shear layer formation in MHD turbulence subject to a guide field and misaligned rotation" Benavides et al., Journal of Fluid Mechanics, 2022. [doi]
  • "eigentools: A Python package for studying differential eigenvalue problems with an emphasis on robustness"
    Oishi et al., Journal of Open Source Software, 2021. [doi]
  • "Improving accuracy of volume penalised fluid-solid interactions"
    Hester et al., Journal of Computational Physics, 2021. [doi]
  • "Improved phase-field models of melting and dissolution in multi-component flows"
    Hester et al., Proceedings of the Royal Society A, 2020. [doi]
  • "Single-hemisphere Dynamos in M-dwarf Stars"
    Brown et al., Astrophysical Journal Letters, 2020. [doi]
  • "Linearly forced fluid flow on a rotating sphere"
    Supekar et al., Journal of Fluid Mechanics, 2020. [ads] [doi]
  • "Dedalus: A Flexible Framework for Numerical Simulations with Spectral Methods"
    Burns et al., Physical Review Research, 2020. [ads] [doi]
  • "The magnetorotational instability prefers three dimensions"
    Oishi et al., Proceedings of the Royal Society A, 2020. [doi]
  • "Low-frequency Variability in Massive Stars: Core Generation or Surface Phenomenon?"
    Lecoanet et al., Astrophysical Journal Letters, 2019. [ads] [doi]
  • "Quantum hydrodynamics for supersolid crystals and quasicrystals"
    Heinonen et al., Physical Review A, 2019. [ads] [doi]
  • "The 'Sphered Cube': A New Method for the Solution of Partial Differential Equations in Cubical Geometry"
    Burns et al., arXiv.org, 2019. [arxiv]
  • "Tensor calculus in spherical coordinates using Jacobi polynomials. Part-II: Implementation and examples"
    Lecoanet et al., Journal of Computational Physics, 2019. [doi]
  • Tensor calculus in spherical coordinates using Jacobi polynomials. Part-I: Mathematical analysis and derivations"
    Vasil et al., Journal of Computational Physics, 2019. [doi]
  • "Anomalous Chained Turbulence in Actively Driven Flows on Spheres"
    Mickelin et al., Physical Review Letters, 2018. [ads] [doi]
  • "Rolling resistance of shallow granular deformation"
    Burns et al., Proceedings of the Royal Society A, 2017. [ads] [doi]
  • "Conversion of Internal Gravity Waves into Magnetic Waves"
    Lecoanet et al., MNRAS, 2016. [ads] [doi]
  • "Turbulent Chemical Diffusion in Convectively Bounded Carbon Flames"
    Lecoanet et al., Astrophysical Journal, 2016. [ads] [doi]
  • "Tensor calculus in polar coordinates using Jacobi polynomials"
    Vasil et al., Journal of Computational Physics, 2016. [ads] [doi]
  • "A validated nonlinear Kelvin-Helmholtz benchmark for numerical hydrodynamics"
    Lecoanet et al., MNRAS, 2016. [ads] [doi]
  • "Numerical simulations of internal wave generation by convection in water"
    Lecoanet et al., Physical Review E, 2015. [ads] [doi]
  • "Conduction in Low Mach Number Flows. I. Linear and Weakly Nonlinear Regimes"
    Lecoanet et al., Astrophysical Journal, 2014. [ads] [doi]
  • "FIRST, a fibered aperture masking instrument. I. First on-sky test results"
    Huby et al., Astronomy & Astrophysics, 2012. [ads] [doi]

Other works

  • "Flexible Spectral Algorithms for Simulating Astrophysical and Geophysical Flows"
    Burns, Doctoral Thesis, 2018. [pdf]
  • "Perspectives on Reproducibility and Sustainability of Open-Source Scientific Software from Seven Years of the Dedalus Project"
    Oishi et al., arXiv.org, 2018. [arxiv]
  • "Chebyshev Spectral Methods with Applications to Astrophysical Fluid Dynamics"
    Burns, Cambridge Part III Essay, 2013. [pdf]