Fellow members of the mathematics department,

There will be a virtual CAM seminar 

Today @ 3:35pm

via Zoom: 

https://tennessee.zoom.us/j/95658794984

The speaker info, title, and abstract are below.

Best,
Abner

Speaker: Ignacio Tomas
Affiliation: Sandia National Laboratories


Title: From compressible Euler to compressible Navier-Stokes: numerical
schemes with mathematically guaranteed properties


Abstract: The first step in the development of a high-order accurate
scheme for hyperbolic systems of conservation laws is the development
of a robust first-order method supported by a rigorous mathematical
basis. With that goal in mind, we develop a general framework of first-
order fully-discrete numerical schemes that are guaranteed to preserve
every convex invariant of the hyperbolic system and satisfy every
entropy inequality.
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: nonlinear
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 are 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 either 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.

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