Research
Our group specializes in the advancement of asymptotic analysis techniques, applied to the study of the long-time behavior of waves in fluids as well as the analytic continuation of divergent or truncated series that arise in areas of mathematical physics such as fluid dynamics, thermodynamics, and astrophysics. These two areas are described below.
Wave Instability
The above animation shows a convectively unstable wave emanating from an oscillating point source. To 3-D print the final frame, visit the 3D models section.
Asymptotic Approximants
Power series arise in virtually all applications of mathematical physics. However, limitations generally inherent to power series solutions often inhibit their direct use. For instance, a Taylor series representation of an unknown function may not converge, as it may have a finite radius of convergence arising from singularities (often complex) in the function it represents. Even when singularities are not a concern, higher-order terms of the series may be exceedingly difficult to compute, which is especially problematic if the series converges slowly. While “re-summation” methods such as Pade Approximants typically lead to an implementation improvement compared with the original series, global accuracy is not always guaranteed and the problem becomes one of choosing the ‘ best’ re-summation technique.
Our goal is to identify relevant problems that can be solved using an asymptotic approximant: a function whose Taylor expansion (about x=ax=a) matches the exact Taylor expansion to some order and whose limit as x→bx→b matches some known asymptotic behavior. Divergent, truncated, and/or slowly converging series can be replaced by asymptotic approximants if some behavior is known away from the series expansion point. Nonlinear ODEs that require the "shooting method" are well suited for asymptotic approximants, applied to the (typically divergent) power series solution to the ODE. So far, we have successfully applied asymptotic approximants to find closed-form analytic solutions to problems in thermodynamics, fluid mechanics, and astrophysics.
Below are 4 examples of how asymptotic approximants (denoted A in the figures) can be used to bridge the behavior between two physical regions. In all figures, symbols represent numerical solutions. top left: 2nd, 4th, and 6th order approximants, whose zero-density expansion matches the square-well virial series (V) to 2nd, 4th, and 6th order, while also limiting to non-classical scaling behavior near the thermodynamic critical point (Barlow et al, JCP, 2015). top right: Approximant matches known weak (W)- and strong (S)-field limits for the bending angle of a Kerr black hole (Barlow, Weinstein, & Faber, CQG, 2017). bottom left: Approximant bridges the series solution (S) of the Blasius ODE (describing boundary layer flow over a plate) with its far-field behavior (Barlow et. al., QJMAM, 2017). bottom right:Approximant bridges the series solution of the Flierl-Petviashvilli ODE (describing vortex solitons like Jupiter's red spot) with the far field behavior (Barlow et. al., QJMAM, 2017).
Beachley, Ryne J, Morgan Mistysyn, Joshua A Faber, Steven J Weinstein, and Nathaniel S Barlow. 2018. “Accurate Closed-Form Trajectories Of Light Around A Kerr Black Hole Using Asymptotic Approximants”. Class. Quantum Grav. 35 (20).
Barlow, Nathaniel S, Steven J Weinstein, and Joshua A Faber. 2017. “An Asymptotically Consistent Approximant For The Equatorial Bending Angle Of Light Due To Kerr Black Holes”. Class. Quantum Grav. 34 (135017).
Barlow, Nathaniel S, Christopher R Stanton, Nicole Hill, Steven J Weinstein, and Allyssa G Cio. 2017. “On The Summation Of Divergent, Truncated, And Underspecified Power Series Via Asymptotic Approximants”. Quarterly Journal Of Mechanics And Applied Mathematics 70 (1): 21-48.
Barlow, Nathaniel S, Andrew J Schultz, Steven J Weinstein, and David A Kofke. 2015. “Communication: Analytic Continuation Of The Virial Series Through The Critical Point Using Parametric Approximants”. The Journal Of Chemical Physics 143 (071103).
Barlow, Nathaniel S, Andrew J Schultz, David A Kofke, and Steven J Weinstein. 2014. “Critical Isotherms From Virial Series Using Asymptotically Consistent Approximants”. Aiche Journal 60l: 3336–3349.
Barlow, Nathaniel S, A. J Schultz, S. J Weinstein, and D. A Kofke. 2012. “An Asymptotically Consistent Approximant Method With Application To Soft- And Hard-Sphere Fluids”. J. Chem. Phys. 137: 204102.
3D Models
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Want to print 3D images? Instructions and pictures have been provided by Arden Bonzo on how to use the printer.
Convectively unstable wave due to an oscillating point source.
Exponentially growing absolute instability (dimension into the page is time).
Algebraically decaying wave (dimension into the page is time).
For more info on the PDE for the two models above, see: Barlow, Nathaniel S, B. T Helenbrook, S. P Lin, and S. J Weinstein. 2010. “An Interpretation Of Absolutely And Convectively Unstable Waves Using Series Solutions”. Wave Motion 47: 564-582.
Algebraically growing absolute instability (dimension into the page is time).
For more info on the PDE for the above model, see: King, Kristina R, Steven J Weinstein, Paula M Zaretzky, Michael Cromer, and Nathaniel S Barlow. 2016. “Stability Of Algebraically Unstable Dispersive Flows”. Phys. Rev. Fluids 1 (7).
The pressure-temperature-density diagram of a square-well fluid at its critical region and above, showing the correct non-classical scaling from all directions.
The equation of state used to produce the above model is given in: Barlow, Nathaniel S, Andrew J Schultz, Steven J Weinstein, and David A Kofke. 2015. “Communication: Analytic Continuation Of The Virial Series Through The Critical Point Using Parametric Approximants”. The Journal Of Chemical Physics 143 (071103).