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.