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A variational multiscale stabilized finite element method for the solution of the Euler equations of nonhydrostatic stratified flows

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A Variational Multiscale Stabilized Finite Element Method for the Solutionof the Euler Equations of Nonhydrostatic Stratiﬁed Flows
Simone Marras
a,
∗
, Margarida Moragues
a
, Mariano Vázquez
a,b,
∗∗
, Oriol Jorba
a
, GuillaumeHouzeaux
a
a
Barcelona Supercomputing Center, Spain
b
IIIA-CSIC. Barcelona, Spain
Abstract
We present a compressible version of the Variational Multiscale Stabilization (VMS) method ap-plied to the Finite Element (FE) solution of the Euler equations for nonhydrostatic stratiﬁed ﬂows.This paper is meant to verify how the algorithm performs when solving problems in the frame-work of nonhydrostatic atmospheric dynamics. This eﬀort is justiﬁed by the previously observedgood performance of VMS and by the advantages that a compact Galerkin formulation oﬀers onmassively parallel architectures —a paradigm for both Computational Fluid Dynamics (CFD) andNumerical Weather Prediction (NWP) practitioners. We also propose a simple technique to con-struct a well-balanced approximation of the dominant hydrostatics that, if not properly discretized,may cause unacceptable vertical oscillations. This is a relevant problem in NWP, especially in theproximity of steep topography. To evaluate the performance of the method for stratiﬁed environ-ments, six standard 2D and two 3D test cases are selected. Of these, two admit a semi-analyticsolution, while the remaining six are non-steady and non-linear thermal problems with dominantbuoyancy eﬀects that challenge the algorithm in terms of stability.
Keywords:
Variational Multiscale Stabilization; Finite Element Method; Nonhydrostatic Flows;Euler Equations.
∗
simone.marras@bsc.es
∗∗
mariano.vazquez@bsc.es
Preprint submitted to Journal of Computational Physics September 10, 2012
*ManuscriptClick here to view linked References
1. Introduction
The past decade has seen an important growth in the development of faster and cheaper super-computers for high-performance computing (HPC). This trend is such that many research groupscan now enhance their models in terms of eﬃciency and accuracy through higher resolution, by lim-iting the time of computation. Among many ﬁelds of computational mechanics, this is even moreso in the context of atmospheric simulations, where the wall-clock time is a discriminant metricin the selection of the underlying algorithm. It has been widely proved that, to eﬀectively see animprovement in computation on these architectures, the numerical algorithm at hand should havecertain characteristics. Of these, being local by construction, which implies the smallness of thecommunication foot-print, is maybe the most important. On these grounds, we propose a FiniteElement (FE) scheme with Variational Multiscale Stabilization (VMS) for the solution of the fullycompressible Euler equations of stratiﬁed ﬂows, having it proved great parallelization eﬃciency onmassively parallel architectures [1]. The use of Finite Element schemes in atmospheric simulationsbegan four decades ago with [2] and [3] in the 60s, continued in the 70s (e.g.[4, 5]), and was followed
by an extensive production of articles in the 80s and 90s with, e.g., [6, 7, 8], who set the foun-
dations of the operational
Global Environmental Multiscale
(GEM) model [9, 10] of the Canadian
Meteorological Center & Meteorological Research Branch (CMC-MRB). In the UK, [11] used ﬁniteelements for the vertical discretization of a semi-Lagrangian transport scheme and introduced it inthe operational version of the European Centre for Medium-Range Weather Forecasts (ECMWF)global spectral model, with great improvement with respect to the ﬁnite diﬀerence (FD) versionof the code. At the same time, in [12] the British Met Oﬃce committed to a FE approach forthe treatment of the vertical atmosphere within their global and regional models. More Galerkin-type models appear since the beginning of 2000 in the domain of Geophysical ﬂuid dynamics: in[13, 14, 15, 16, 17, 18], diﬀerent variational formulations mostly based on Spectral Elements (SE)
and Discontinuous Galerkin (DG) techniques are employed to solve the shallow water problem,hyperbolic systems on the sphere, or the Navier-Stokes and Euler equations in non-hydrostaticmesoscale and global modeling with unstructured grids. Similarly and with dynamic mesh adap-tivity in mind, Numerical Weather Prediction (NWP) Finite Volume (FV) softwares such as theJapanese
Nonhydrostatic Icosahedral Atmospheric Model
(NICAM) on global domains ([19, 20])
and the American
Operational Multiscale Environmental model with Grid Adaptivity
(OMEGA)2
[21] are further examples of new trends in NWP as alternatives to FD methods. More recently,examples of element-based models are the SE/DG
Nonhydrsotatic Uniﬁed Model for the Atmo-sphere
(NUMA), whose linear scalability up to 12,288 CPUs using MPI was shown in [22], and the
SE
Community Earth System Model
(CESM) with scalability shown up to 160,000 CPUs in [23].
Müller et al. are working on adaptive grid reﬁnement and built an adaptive solver for atmosphericproblems proposed in [24]. In 2010, [25] presented their results from an edge-based ﬁnite element
solver built on a fully unstructured grid. In this context, the ﬁnite element algorithm proposed hereis, at our knowledge, the ﬁrst continuous Galerkin method with VMS stabilization applied to strat-iﬁed nonhydrostatic ﬂows. No special numerical treatment or special assumption on the governingequations are considered for the low Mach regime typical of atmospheric dynamics. Nevertheless,this algorithm can treat a widespread range of ﬂow regimes including very low Mach number ﬂows.When convection dominates, because of the centered nature of a straight ﬁnite element dis-cretization instabilities form until the solution eventually explodes. These instabilities can betreated by means of diﬀerent stabilization procedures. The
Streamline Upwind Petrov-Galerkin
(SUPG) [26, 27], the
Galerkin/Least-Squares
(GLS) [28], Galerkin methods with bubble functions
[29, 30, 31], or sub-grid projection methods [32] are some of the most used stabilization techniques
for ﬁnite elements. Due to its high eﬃciency, robustness, and validity at all Mach regimes, in thispaper we focus on the Variational Multiscale approach pioneered by [33, 34]. To our knowledge,
VMS methods for compressible ﬂows today only appear in [35, 36, 37, 38] for Computational Fluid
Dynamics (CFD) simulations where compressibility eﬀects are large (i.e.
M
≥
0
.
36, where
M
indicates the ﬂow Mach number). In the framework of atmospheric modeling, VMS was recentlyapplied by [39] to solve the advection-diﬀusion equation using high-order spectral elements. Here
we use a simpliﬁed version of the method presented in [38].The main advantages of the method can be summarized as follows. The algorithm is halo-free,which is one of the biggest advantages for eﬃcient parallelization regardless of the order of accuracyof the scheme [16]. The only information that is needed by each element is that of the shared
nodes between neighboring elements. This makes the method fully local and highly parallelizable.Although quadrilateral elements are used, the algorithm is fully unstructured and does not relyon speciﬁc characteristics of the grid. Furthermore, no dimensional splitting is used in that boththe horizontal and vertical directions are not distinguished in the discretization process. In other3
words, the Euler equations are solved by the same numerical method in all space dimensions. Thegreat advantage of this is that the code becomes totally free from the geometry of the grid. Thisapproach is classical in CFD and was ﬁrst applied in NWP by [21, 40, 41] using ﬁnite volumes,
ﬁnite diﬀerences, and high order spectral elements, respectively.The paper is organized as follows. Section 2 describes the set of equations used throughout.The numerical method of solution is described in Section 3 and is followed by a description of well-balanced solutions in Section 4. In Sections 5 and 6 the algorithm is tested against standard
benchmarks for atmospheric simulation in two and three dimensions respectively. Conclusions arereported in Section 7.
2. Governing equations
Let
d
= 3 be the space dimension. Given a bounded domain Ω
⊂
R
d
and a time interval (0
,t
f
),
t
f
∈
R
+
, we write the Euler equations of atmospheric ﬂows [42] with no Coriolis eﬀects as:
∂
q
∂t
+
L
(
q
) =
f
(
q
)
,
(1)where
q
is the vector of the unknowns,
L
is the diﬀerential operator, and
f
is the source vector.We have:
q
=
ρU V W θ
,
L
(
q
) =
∇·
U
∇·
U
U ρ
+
p
e
x
∇·
U
V ρ
+
p
e
y
∇·
U
W ρ
+
p
e
z
U
ρ
·∇
θ
,
f
(
q
) =
000
−
ρg
0
,
where
∇
=
∂ ∂x
,
∂ ∂y
,
∂ ∂z
, and
e
x
= (1
,
0
,
0)
T
,
e
y
= (0
,
1
,
0)
T
, and
e
z
= (0
,
0
,
1)
T
are the unitaryvectors in the space directions
x
,
y
, and
z
, respectively. Superscript T indicates the transposeoperator. Density
ρ
, momentum
U
= (
U,V,W
)
T
, and potential temperature
θ
are functions of space
x
= (
x,y,z
) and time
t
. We also deﬁne the velocity
u
=
U
/ρ
= (
u,v,w
). The acceleration4
of gravity, of modulus
g
, acts along the vertical direction. It is understood that in two dimensions,the
y
-direction disappears. System (1) is closed by the state law for pressure
p
=
p
0
Rρθ p
0
γ
,
(2)where
γ
=
c
p
/c
v
, being
c
p
and
c
v
the coeﬃcients of speciﬁc heat at constant pressure and volume,respectively,
R
=
c
p
−
c
v
is the constant of perfect gases, and
p
0
= 10
5
Pa
is the surface pres-sure. The problem consists in ﬁnding
q
(
x
,t) such that Eq. (1) with proper boundary and initialconditions is veriﬁed
∀
(
x
,t
)
∈
Ω
×
(0
,t
f
).
3. Numerical formulation
Set (1) is discretized in space using Finite Elements stabilized by the Variational Multiscalemethod and in time using an explicit Finite Diﬀerence scheme.
3.1. Finite Elements: Weak Form
Given a polyhedral approximation Ω
h
of Ω, let
P
h
=
{
K
i
}
i
=1
,...,n
el
be its ﬁnite element partition,where
K
i
⊂
Ω
h
are
n
el
conforming quadrilaterals of characteristic length
h
i
. In this paper,
h
i
corresponds to the shortest edge of the element. We consider the trial and basis functions space
W
h
associated with
P
h
as the space generated by the Lagrange polynomials of order one. Let
{
p
k
}
k
=1
,...,n
nodes
be the nodes of the grid and
ψ
hk
the Lagrange polynomial associated with node
p
k
,then
{
ψ
hk
}
k
=1
,...,n
nodes
is a basis for
W
h
. We project Eq. (1) onto
W
h
by the
L
2
scalar product,where
L
2
is the space of square-integrable real-valued functions. We hence obtain the weak (orvariational) form of Eq. (1):
Ω
h
ψ
h
∂
q
∂t d
Ω
h
+
Ω
h
ψ
h
L
(
q
)
d
Ω
h
=
Ω
h
ψ
h
f
(
q
)
d
Ω
h
∀
ψ
h
∈
W
h
.
(3)We deﬁne
q
h
as the projection of
q
onto
W
h
and expands it as
q
h
(
x
,t
) =
n
nodes
k
=1
ψ
hk
(
x
)
q
hk
(
t
)
,
(4)5

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