User:Tohline/PGE/Hybrid Scheme Preface
Hybrid Advection Scheme (Preface)
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March 1, 2014 by Joel E. Tohline
Throughout my research career, I have sought new or modified techniques and algorithms that would allow my group to perform more accurate numerical simulations of astrophysical fluid flows. At the beginning — following the advice of my dissertation co-advisors, Peter Bodenheimer and David Black — I adopted a cylindrical, rather than Cartesian, computational grid. When a cylindrical coordinate system is used, one of the components of the equation of motion can be written in a form that fairly naturally conserves angular momentum, and this was quite a desirable feature, given that our investigations were focusing on analyzing the stability of rotating configurations.
During the mid-90s, when Kimberly Barker New was conducting her dissertation research (see New & Tohline 1997), we started using a rotating cylindrical coordinate mesh. On a grid that was spinning at a suitably chosen frequency, advection of fluid through the grid could be minimized and this, in turn, reduced the undesirable effects of numerical diffusion. We adopted a fairly standard algorithmic approach very similar to the one used by Norman & Wilson (1978) and acknowledged that we were making a tradeoff: While the shift to a rotating cylindrical coordinate mesh reduced the effects of numerical diffusion, the shift introduced a rather ugly Coriolis "source" term into two components of the equation of motion. This made it more difficult to ensure conservation of angular momentum.
One day I noticed that, while imposing some fairly reasonable constraints, the Coriolis term could be removed from the "source" term and folded into the divergence term on the left-hand side of the angular momentum conservation equation. This manipulation of terms seemed to be saying that the undesirable Coriolis term would disappear while employing a rotating coordinate mesh if the variable that was advected through the grid was the inertial-frame angular momentum density, rather than the rotating-frame angular momentum density. This seemed too good to be true. The discovered code modification would allow us to conserve angular momentum very accurately and, at the same time, allow us to use a rotating grid and thereby minimize numerical diffusion. My early notes on this topic have been preserved, as they were included in my earliest version of this web-based H_Book; the relevant page can be accessed here, which is an html file whose linux time stamp is August 27, 2000. The symbol fonts utilized throughout this old html page seem now to be readable only through Microsoft's Internet Explorer web browser. Hence, for posterity sake, I have retyped this "year 2000" set of notes into an accompanying page of this wiki. As the notes indicate, my group began referring to this as the "A* scheme".
We delayed implementing this A* advection scheme in our production code for a number of years, primarily because it was unclear to me how to derive — and, therefore, fully justify — this hybrid inertial/rotating-frame advection scheme in full three-dimensional generality. How was the Coriolis term in the radial component of the equation of motion to be concurrently handled, for example? Jay Call's dissertation research focused precisely on this question (see Call, Tohline, & Lehner 2010). He derived a complete description of the hybrid advection scheme in a fully relativistic and generalized coordinate framework. Jay showed that it is indeed valid to advect inertial-frame quantities across a rotating grid, in the manner suggested by my simpler A* scheme derivation. In addition — and more importantly — he showed how to write the system of fluid equations to allow advection of inertial-frame angular momentum (generally associated with a cylindrical coordinate mesh) across a rotating Cartesian grid.
July 11, 2015 by Joel E. Tohline
Primarily in the context of Zach Byerly's doctoral dissertation research, my group has since demonstrated that this hybrid advection scheme works extremely well. The key refereed publication resulting from this work is Byerly, Adelstein-Lelbach, Tohline, & Marcello (2014).
Go to: Hybrid Advection Scheme
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