Georgia Institute of Technology College of Sciences

To treat the multiple time scales of ocean dynamics in an efficient manner, the baroclinic-barotropic splitting technique has been widely used for solving the primitive equations for ocean modeling. In this paper, we propose second and third-order multirate explicit time-stepping schemes for such split systems based on the strong stability-preserving Runge-Kutta (SSPRK) framework. Our method allows for a large time step to be used for advancing the three-dimensional (slow) baroclinic mode and a small time step for the two-dimensional (fast) barotropic mode, so that each of the two mode solves only need satisfy their respective CFL condition to maintain numerical stability. It is well known that the SSPRK method achieves high-order temporal accuracy by utilizing a convex combination of forward-Euler steps. At each time step of our method, the baroclinic velocity is first computed by using the SSPRK scheme to advance the baroclinic-barotropic system with the large time step, then the barotropic velocity is specially corrected by using the same SSPRK scheme with the small time step to advance the barotropic subsystem with a barotropic forcing interpolated based on values from the preceding baroclinic solves. Finally, the fluid thickness and the sea surface height perturbation is updated by coupling the predicted baroclinic and barotropic velocities. Two benchmark tests drawn from the ``MPAS-Ocean" platform are used to numerically demonstrate the accuracy and parallel performance of the proposed schemes.

The bluejeans link for the seminar is https://bluejeans.com/457724603/4379

Donaldson’s Diagonalization Theorem has been used extensively over the past 15 years as an obstructive tool. For example, it has been used to obstruct: rational homology 3-spheres from bounding rational homology 4-balls; knots from being (smoothly) slice; and knots from bounding (smooth) Mobius bands in the 4-ball. In this multi-part series, we will see how this obstruction works, while getting into the weeds with concrete calculations that are usually swept under the rug during research talks.

This talk is based on a chapter that I wrote for a book in honor of John Benedetto's 80th birthday.  Years ago, John wrote a text "Real Variable and Integration", published in 1976.  This was not the text that I first learned real analysis from, but it became an important reference for me.  A later revision and expansion by John and Wojtek Czaja appeared in 2009.  Eventually, I wrote my own real analysis text, aimed at students taking their first course in measure theory.  My goal was that each proof was to be both rigorous and enlightening.  I failed (in the chapters on differentiation and absolute continuity).  I will discuss the real analysis theorem whose proof I find the most difficult and unenlightening.  But I will also present the Banach-Zaretsky Theorem, which I first learned from John's text.  This is an elegant but often overlooked result, and by using it I (re)discovered enlightening proofs of theorems whose standard proofs are technical and difficult.  This talk will be a tour of the absolutely fundamental concept of absolute continuity from the viewpoint of the Banach-Zaretsky Theorem.

Zoom Link:  https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09