Multiresolution analysis & simulation using spectral elements
(MASUSE)
Aimé Fournier
October 7, 2006
Turbulence Numerics Team – 2006 NAR
in collaboration with Duane Rosenberg & Annick Pouquet
Sponsored by NSF (NCAR’s Programs)
1
Historical and scientific context
From a mathematical point of view the fundamental challenge of computational fluid dynamics
arises from nonlinear terms in the governing dynamical equations, of which the prototype is the
~ u. It is commonly known that the most accurate and efficient
flow velocity ~u and its advection (~u · ∇)~
~ part is the spectral method, in which ~u(~x) is approximated at
method to compute the gradient (∇)
~
each point ~x by a series in coefficients ~u~k of a set of smooth global basis functions such as {eik·~x }~k∈K ;
however the most efficient method (especially in massive parallel computation) for the multiplication
part (·) is by a set {~u(~x~)}~∈J of values corresponding to localized basis functions such as the finite
elements e.g., the continuous function φ~ı(~x) which equals δ~ı,~ at ~x = ~x~ and has uniform gradient
inside certain disjoint simplexes that have vertices at the ~x~ and cover the domain D. Unless the D
geometry is trivial there is no known fast transform between the ~u~k and ~u(~x~), so some compromise,
intermediate representation would be very desirable. One that combines the efficiency and geometric
flexibility of finite elements with the accuracy of spectral methods is the spectral-element method
[SEM, 16]. SEM was introduced to NCAR with application to both shallow-water [22] and 3D
hydrostatic [9, 21, 23] dynamics on the sphere.
~ u is that they can lead to strongly localized multiscale features
The challenge of terms like (~u · ∇)~
such as fronts, plumes and vortices, whose accurate description and prediction in regimes of realistically high Reynolds number (R, typical ratio of inertial to viscous force) exceeds the capability of
today’s supercomputers if uniform, static meshes are used; this motivates the development of SEM
with dynamically adaptive mesh refinement (AMR). Dynamic AMR-SEMs have been formulated
for 1D [14] and 2D spherical [20] and planar [7, 11, 15] fluid-dynamics computations, in particular,
GASpAR1 [18]. Fournier [7] helped build connections between the methods of dynamic AMR-SEM
and wavelets for PDEs; where the history of the latter spans 100s of reports for at least 17 years2
[e.g., 1, 2, 17]. The methods described in the present report may be broadly labeled multiresolution
analysis & simulation using spectral elements, or MASUSE. Multiresolution analysis generalizes
Fourier analysis and is related to wavelet analysis, where the latter has been applied by NCAR
scientists regularly for at least 12 years (at least 33 NCAR citations from [10] to [13]).
1
2
“GASpAR development” http://www.cisl.ucar.edu/nar/2006/catalog/Rosenberg1.pdf
Web of Science 2006/10/7 returns 352 documents with topic wavelet* and ("differential eq*" or pde*)
1
2
How MASUSE supports NCAR strategic priorities
MASUSE supports various NCAR/CISL research efforts,3 especially GASpAR and other adaptive
models4 as well as providing new tools.5 In particular it supports applications to turbulence simulation and modeling.6 The 2006 NCAR Strategic Plan 7 observes that “developing an integrated
understanding of the Earth system, including everything from human influences to the role of the
Sun, is the most important task facing the atmospheric and related sciences. Such an understanding
requires observing and predicting the behavior of large complex systems operating at multiple scales
of time and space.” With this motive, MASUSE is one of the the “new studies that integrate and
synthesize classical subdisciplines, permitting analysis of critically important interactions across
the spatial and temporal scales of natural processes” and aims towards “improved understanding
of atmospheric processes across scales”, to “provide robust, accessible, and innovative information
services and tools” and to “create a conceptual framework for integrating research across time and
space scales to aid decision makers and enrich understanding of processes across scales”. A long-term
application would be to remedy that “current frameworks for climate modeling do not fully represent the upscale influence of mesoscale processes and systems” by enabling “integrated regional-scale
investigations of climate and weather impacts”. More generally, multiresolution methods are vital
to recent and ongoing IMAGe8 and CISL9 activities.
3
2006 accomplishments
Two major accomplishments completed in fy2006 were the invention of new algorithms for continuous, nonconforming spectral elements, to compute both exact global Fourier coefficients ~u~k [5] and
a new multiresolution analysis (MRA) type [6]. As an illustration of the former, Fig. 1 shows a
recent GASpAR [18] dynamically adaptive simulation of a 2D incompressible Navier-Stokes flow of
a challenging test problem in the literature [3, 12, 19]. It also shows the kinetic-energy spectrum
computed using the new, exact transform [5]. The presence of different wavenumber-scaling regimes
corresponding to vortex cores and filaments at different times is suggested.
3
see Applied Mathematics http://www.cisl.ucar.edu/research/math.jsp and the 2005 Annual Report http:
//www.nar.ucar.edu/cyber4.jsp
4
see Models and Software Frameworks http://www.cisl.ucar.edu/research/models.jsp, AMR for Weather &
Climate http://www.cisl.ucar.edu/css/staff/cjablono/amr.html, and especially TNT Plans http://www.cisl.
ucar.edu/nar/2005/res/tnt.p.jsp
5
see CISL Plans http://www.cisl.ucar.edu/nar/2005/res/crat.p.jsp and the 2005 User Forum http://www.
cisl.ucar.edu/news/05/lead/0527.userforum.html
6
see 2005 accomplishments http://www.cisl.ucar.edu/nar/2005/res/tnt.a.jsp
7
http://www.ncar.ucar.edu/stratplan/2006/strategy rnd2 v5.pdf
8
see GSP Plans http://www.cisl.ucar.edu/nar/2005/res/gsp.p.jsp and Theme-of-the-Year 2006 http://www.
cisl.ucar.edu/nar/2005/res/toy.p.jsp
9
see VAPOR http://www.cisl.ucar.edu/news/05/lead/1109.abstracts.jsp
2
Figure 1: Left-hand side shows a vorticity-field snapshot of numerically simulated [using GASpAR,
18] 2D turbulent decay of a 3-vortex initial state at R = 2 × 104 at 4.0 eddy turnover times, with 5
octaves of element sizes, and (7, 7)-degree polynomial expansion in each element. Right-hand side
shows the kinetic-energy spectrum history (computed directly on non-conforming grids as shown at
left [5]), from red at present extending back to the log-parabolic initial state. An animation of this
figure is at http://www.image.ucar.edu/∼fournier/projects/nug3vort/.
3
To outline the new MRA, recall that the SEM partitions D into disjoint elements D̄ =
SI
i=1 Ēi ,
that each support polynomial expansions along coordinates xα in a basis {φi,~(~x)} that interpolates
the mapped nodes {~xi,~}. The extra local degrees of freedom (d.o.f.) per element provide both high
accuracy and a local accuracy estimate. As explained elsewhere [8], an effective adaptivity criterion
is enabled by the multiresolution hierarchy intrinsic to SEM, i.e.,
span
[
φi,~ =: Pi (
~∈×dα=1 {0,...pα }
S2d (i+1)
ı̃=2d i+1
Pı̃
[
⇒
Pi =: V` ( V`+1 ,
(1)
(`+1)d
`d
i∈{ 2 d −1 ,... 2 d −1 −1}
2 −1
2 −1
Eı̃ =Ei
as introduced in detail by [6]. The projection operator ℘` := projV` \V`−1 on fine scales is the
continuous spectral-element analog to the discontinuous spectral-element projection [7], but is not
orthonormal; nevertheless, it allows exact discrete-L2 decomposition. These projections ℘` are
rigorous analogs for continuous, nonconforming spectral elements, of Fourier spectral wavenumber
bandpass filters
F ` ~u(~x) :=
X
~
F~k` u~k eik·~x ,
d
supp F ` ≈ {~k; 2` + 1 ≤ max |k α | ≤ 2`+1 }
α=1
~k∈Zd
(2)
on uniform grids; therefore they can enable new methods which have potential application in Turbulence Numerics Team research, such as spatially localized triadic interactions between scales 10
[4], filter-based modeling of the dynamics,11 and automated extraction of coherent or intermittent
structures.12
The criterion to dynamically adapt the computation mesh at a time t in regard to merging 2d
neighbor elements ı̃ into one element i starts by examining the relative contributions of
{~u(t, ~xi,~)}
and {~u(t, ~xı̃,~)}ı̃∈{2d i+1,...2d (i+1)}
to the local discrete L2 norm of the solution piecewise-polynomial ~u(t, ·), restricted to Ei . This
criterion has been applied to adaptive representation [5].
In Fig. 2, the multiresolution analysis based on (1) is applied to simulation by GASpAR [18] of
the same flow as Fig. 1 but using a fixed 32 × 32 elements (each of 2D polynomial degree (16, 16)).
Initially only large-scale modes are present because the vorticity is merely a sum of 3 Gaussians,
but over time there appear ever smaller-scale modes. This new MRA [6] shows that an entirely local
10
“Effects of large scales on the statistics and dynamics of turbulent flows” http://www.cisl.ucar.edu/nar/2006/
catalog/Alexakis1.pdf, “Inverse cascade of magnetic helicity” http://www.cisl.ucar.edu/nar/2006/catalog/
Alexakis3.pdf and “Inverse cascades and α-effect at low magnetic Prandtl number” http://www.cisl.ucar.edu/
nar/2006/catalog/Mininni1.pdf
11
“Modeling of turbulent flows” http://www.cisl.ucar.edu/nar/2006/catalog/Graham1.pdf
12
“Hydrodynamics and magnetohydrodynamics in rotating spheres” http://www.cisl.ucar.edu/nar/2006/
catalog/Mininni2.pdf and “Small scale structures in MHD flows” http://www.cisl.ucar.edu/nar/2006/catalog/
Pouquet1.pdf
4
analysis intrinsic to the spectral-element computational representation (middle) provides essentially
the same orthogonal scale partitioning as the traditional but more computationally demanding
global analysis (bottom).
4
2007 plans
In 2007 we intend to:
• Fully implement the new adaptivity criterion based on (1) into GASpAR;
• Quantitatively evaluate the effectiveness of MASUSE for strongly multiscale problems;
• Quantitatively evaluate the effectiveness of the new local Fourier transform [5];
• With these new tools address such scientific questions as, what advantages does dynamic
AMR offer to compute 2D high-R flows with and without non-trivial boundary conditions,
background flow, buoyancy, topography, or a free surface? Can effectively higher R and other
challenging parametric regimes be reached with dynamic AMR, and if so, what would be
learned thereby about turbulence?
References
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5
Figure 2: Numerical simulation [using GASpAR, 18] of the 2D incompressible Navier-Stokes equations at Reynolds-number 2 × 104 at 2.6 turn-over times as: top, physical-space vorticity field
ζ (contours log2 (−ζ/π) = −2, . . . 0 & log2 (+ζ/π) = −5, . . . 0); middle, multiresolution analysis
(MRA) using (1) on levels ` = 0, . . . 4 (l to r); and bottom, filtering of global Fourier coefficients (2)
for ` = 0, . . . 4 (l to r). An animation of this figure is at http://www.image.ucar.edu/∼fournier/
projects/nugmasse/.
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