Wednesday, March 3, 2010
Location: ESS, Room 450 2:15PM
V&V for Astrophysical Fluid Simulations
Verification and Validation (V&V) for a numerical simulation asserts two properties. The first (Verification) is that the equations as posed are solved correctly, with controlled numerical errors. The second (Validation) asserts that the equations as solved are an actual description of the physics problem to be solved.
We point out fundamental obstacles to V&V for standard astrophysical codes, and we trace the problems to fundamental issues of a mathematical nature. Then we show how these obstacles can be overcome, leading to a fluid simulation code which has been verified and validated on analogues of astrophysical problems. Finally we show how the specialized numerical routines which have made this advance possible can be combined with any standard astrophysical fluid code, and how a useful fraction of them can be readily inserted directly into any standard astrophysical fluid code.
The test problems for this V&V exercise are analogues of, or details from full fledged astrophysical situations. We study specifically the classical Rayleigh-Taylor (RT) (steady acceleration) and Richtmyer-Meshkov (RM)(shock acceleration) instabilities of a perturbed fluid interface.
A number of startling conclusions are observed. For a flow accelerated by multiple shock waves, we observe an interface between the two fluids proportional to Delta x^-1, that is occupying a constant fraction of the available mesh degrees of freedom. This result suggests
(a) nonconvergence for the unregularize
Mathematical problem or
(b) nonuniqueness of the limit if it exists, or
(c) limiting solutions only in the very weak form of a
space time dependent probability distribution.
The cure for this pathology is a regularized solution, in other words inclusion of all physical regularizing effects, such as viscosity and physical mass diffusion.
In other words, the amount and type of regularization of an
unstable flow is of central importance. Too much regularization, with a
numerical origin, is bad, and too little, with respect to the physics,
is also bad. For systems of equations, the balance of regularization
between the distinct equations in the system is of central importance.
At the level of numerical modeling, the implication from this insight is to compute solutions of the Navier-Stokes, not the Euler equations. Resolution requirements for realistic problems make this solution impractical in most cases. Thus subgrid transport processes must be modeled, and for this we use dynamic models of the turbulence modeling community. In the process we combine and extend ideas of the capturing community (sharp interfaces or numerically steep gradients) with conventional turbulence models, usually applied to problems relatively smooth at a grid level.
This strategy is verified (convergence under mesh refinement)
for a Richtmyer-Meshkov unstable turbulent mixing problem and it is validated(comparison to laboratory experiments) through the study of Rayleigh-Taylor unstable flows. These steps solve problems of some five decades standing in the area of turbulent mixing.
The essential role of a number of collaborators is gratefully