PNNL-13501 - Pacific Northwest National Laboratory

loss of local and global curvature data will introduce large
scaling errors. Also, the uncertainty as to whether a
nonsymmetric result is due to numerical errors or to the
physical continuum model severely limits our predictive
capabilities and our basic understanding of the dynamics
of coupled transport phenomena.
For the linearized equations of mathematical physics,
there exists a well-developed theory of finite difference
schemes. These schemes are based on three fundamental
concepts: 1) truncation error analysis, 2) numerical
stability analysis, and 3) convergence, which follows
from an estimate of the truncation error and the stability
of the numerical scheme. It is possible to extend
truncation error measures to nonlinear equations, but
investigating the numerical stability of methods for
solving such equations is much more complex. A general
mathematical theory of numerical stability for threedimensional
advection-diffusion equations does not
currently exist. The hope of obtaining an n-dimensional
characterization requires that we first determine the
“observability” and identification of the symmetry
invariants of the partial differential equations state
variable representation, given as an n-dimensional
manifold. By carefully examining the parameter space
data of both the time and phase space domains of quasilinear
and nonlinear equations in terms of their linear
equation analogs, we can begin to develop identifying
measures of stability and instability. We will begin to
develop n-dimensional grids to discretize the multiscale,
multidimensional energy surfaces of phase space
representations of these equations. Given that there is
currently no mathematical theory for obtaining
quantitative a priori estimates for the accuracy of finite
difference approximations to full three-dimensional (and
generalized to n-dimensional) nonlinear hydro-transport
equations, we will initially use qualitative
multidimensional grid methods to derive approximate
measures of the time and phase space domains of
multiscale hydro-transport physics domains. Therefore, it
is natural to require that the invariant grid discretized
models should preserve the physical symmetry of the
physical continuum model by maintaining both accurate
measures of local and global curvature, together with the
local algebraic symmetry groups that provide us with a
Lie Group symmetry invariant based definition of
“adequate numerical approximations,” where “adequate”
still needs to be defined.
Preservation of spatial symmetry is one of the
manifestations of the more general notion of invariance of
finite difference methods, related to the preservation of
group properties (up to isometric-isomorphisms) of the
continuum equations and both local and global curvature
symmetry. The theory of Lie Groups, which has been a
classical tool for the analysis of the fundamental
symmetries of continuum ordinary differential equations
for over 100 years, has been used recently to construct
finite difference schemes with symmetry preserving group
invariance in one-dimensional (Vorodnitsyn 1994). A
number of results also are available for three-dimensional
advection-diffusion equations in Eulerian form (Leutoff
et al. 1992). Extending these approaches to two- and
three-dimensional Lagrangian advection-diffusiontransport
equations will lead to significant improvements
in our simulation capabilities.
Results and Accomplishments
The principal investigators have continued to investigate
and apply the Fowler, Oliveira, and Trease (Millennium)
remapping algorithm to three-dimensional particle
transport in a fluid that interacts with a visco-elastic
membrane. (See examples in figures below.) We have
demonstrated that the Millennium algorithm has a nearoptimal
space and time complexity, that is O (n log (n))
and preserves the underlying invariant SO(3) symmetry of
this hybrid grid by faithfully remapping coordinate-free
representations of sets of scalars, vectors, and tensors.
We can show that each of the measure-preserving maps
are symmetry invariant with respect to the SO(3) and the
O(3) groups and that the computational efficiency of the
intersection algorithm depends on the SO(n) invariance of
the spherically bounded search space of polyhedral face
intersections.
The Malard-Oliveira remapping algorithm has a
conjectured computational amortized complexity of O (n).
Significant improvements over the original algorithm
design have been made, which is described as follows:
During the first year of this project, we developed a novel
remapping algorithm, whose expected run-time was
potentially linear in the number of pairs of overlapping
elements. Our second year goal for refining and
implementing the algorithm was to initially develop and
implement an abstract data type for dynamic list
representations that would be more computationally
effective on nonuniform memory access multiprocess
architectures. Figure 1 shows the remapping time for all
three variants. The remapping time excludes both the
input time and the time needed to compute adjacency
relations between elements of the same grid. The runtime
of the bridge variant appears more uniform for large
problems than the other two variants.
A novel remapping algorithm has been simplified,
implemented using abstract data types built on top of
freely available software, and benchmarked on a Silicon
Computational Science and Engineering 133