| Colloquia A tale of the Hilbert matrix Dragan Vukotic Department of Mathematics, Universidad Autónoma de Madrid |
The Hilbert matrix also induces a bounded operator on various other sequence spaces and also on some spaces of analytic functions (including certain Hardy and Bergman spaces), as was noticed from 2000 on. In the second (and more specialized) part of the talk, we will review these more recent developments, including norm computations on different spaces and some more recent results regarding the spectrum of the induced operator.
This Departmental Colloquium is held in conjunction with the Analysis seminar groug, and may be virtually attended via this Zoom link.
| Colloquia From compressible Euler to compressible Navier-Stokes: numerical schemes with mathematically guaranteed properties Ignacio Tomás Center for Computing Research, Sandia National Laboratories |
We then proceed to present a new flux-limiting technique in order to recover second order (or higher) accuracy in space. This technique does not preserve or enforce pointwise bounds on conserved variables, but rather bounds on quasiconvex functionals of the conserved variables. This flux-limiting technique is suitable to preserve pointwise convex constraints of the numerical solution, such as: positivity of the internal energy and minimum principle of the specific entropy in the context of Euler’s equations. Catastrophic failure of the scheme is mathematically impossible. We have coined this technique “convex limiting’’.
Finally, we extend these developments to the case of compressible Navier-Stokes equations using operator-splitting in-time: hyperbolic terms are treated explicitly, parabolic terms are treated implicitly.
Operator-splitting is neither a new idea nor a widely adopted technique for compressible Navier-Stokes equation, most frequently received with skepticism. Contradicting current trends, we developed an operator-splitting scheme for which:
(i) Positivity of density and internal energy can be mathematically guaranteed.
(ii) Implicit stage uses primitive variables, but satisfies a total balance of mechanical energy. This is the key detail that is largely missing in most publications advocating the use of primitive variables and/or operator splitting techniques.
(iii) The scheme runs at the usual "hyperbolic CFL" dt <= O(h) dictated by Euler's subsystem, rather than the technically inapplicable "parabolic CFL" dt <= O(h^2).
The scheme is second-order accurate in space and time and exhibits remarkably robust behavior in the context of shock-viscous-layers interaction. We are not aware of any scheme in the market with comparable computational and mathematical credentials.
This Departmental Colloquium for invited speaker and job candidate Dr. Ignacio Tomás is held in conjunction with the Applied Math seminar group. Please virtually attend via this zoom link.