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

Research Areas


Modern spectral methods for PDEs

Spectral methods are a technique for discretizing PDEs that produce extremely accurate results in simple geometries. My work focuses on developing fast and flexible spectral solvers for large-scale scientific applications. My work in this area includes:

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

Collaborators: Ben Brown, Dan Fortunato, Eric Hester, Daniel Lecoanet, Jeff Oishi, Geoff Vasil
References: Hester et al. (2021), Hester et al. (2020), Burns et al. (2020), Burns et al. (2019), Lecoanet et al. (2019), Vasil et al. (2019), Vasil et al. (2016)


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: Oishi et al. (2021), Burns et al. (2020), Oishi et al. (2018), Lecoanet et al. (2016)


Geophysical & astrophysical fluid dynamics

I have used Dedalus and other numerical tools to study broad ranges of geophysical and astrophysical fluid flows. This work has primarily focused on stellar and planetary waves, turbulence, and dynamos. I am particularly interested in cryospheric processes including ice-ocean interactions and the dynamics of icy moons.

Collaborators: Santiago Benavides, Ben Brown, Glenn Flierl, Eric Hester, Daniel Lecoanet, Jeff Oishi, Geoff Vasil, Nevin Weinberg, Andrew Wells
References: Oishi et al. (2022), Benavides et al. (2022), Benavides et al. (2021), Hester et al. (2020), Brown et al. (2020), Oishi et al. (2020), Lecoanet et al. (2019), Burns (2018), Lecoanet et al. (2016), Lecoanet et al. (2016), Lecoanet et al. (2015)


Complex flows: biological & quantum fluids

The flexibility of the Dedalus framework makes it a powerful tool for simulating complex fluids, including models of biological and quantum systems. Such models can often be simulated at scales which allow for direct comparisons with laboratory data. My collaborators and I are currently using Dedalus to study various models of biophysical processes and active matter. In the quantum domain, I have recently studied wavepacket scattering and supersolid crystals.

Collaborators: Jörn Dunkel, Vili Heinonen, Jonasz Słomka, Nico Romeo
References: Jackson et al. (2022), Heinonen et al. (2022), Supekar et al. (2020), Heinonen et al. (2019), Mickelin et al. (2018)

Current 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 2023
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

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

Other works

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