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Study of a highorder Discontinuous Galerkin method / Finite Differences method
coupled solver on hybrid meshes for CAA
Raphaël LEGER (DSNA/PS3A) – Ph.D Student / 2nd year
Advisor : Serge PIPERNO (CERMICS  ENPC)
(iii) Preliminary results
Bibliography
CONTEXT
(ii) Meshing strategy and coupling algorithm
[1] C.K.W. TAM and J.C. WEBB.
Dispersionrelationpreserving ﬁnite di
ﬀ
erence schemes in computational acoustics.
Journal of Computational Physics,
aug 1993.
[2]
J. UTZMANN, F. LORCHER, M. DUMBSER, and C. MUNZ.
Aeroacoustic simulations for complex
geometries based on hybrid meshes
. In 12th
AIAA/CEAS aeroacoustics conference, may 2006.
[3] B. COCKBURN and C.W. SHU.
Rungekutta discontinuous galerkin methods for convectiondominated problems.
NASA Technical report, nov 2000.
[4] P. DELORME, P.A. MAZET, C. PEYRET, and Y. VENTRIBOUT.
Computational aeroacoustics applications based on a discontinuous galerkin
method.
Comptes rendus de l'académie des sciences, jul 2006.
Background
STRATEGY AND PRELIMINARY RESULTS
(0) Modelling background and implemented methods
(iv) Future developments and analysis
In the ﬁeld of direct numerical simulation of acoustic waves in the presence of obstacles
and/or an heterogeneous ﬂow, both
Discontinuous Galerkin
(DG) methods and
Finite
Diﬀerences
(FD) methods are widely spread. These methods hold speciﬁc
advantages
and drawbacks
. In particular, we recall
FDM
[1] are rather
easy to implement
and show
good diﬀusion and dispersion properties
. On the other hand, they are
not well adapted to
take in account complex geometries
, as they require to be run on structured meshes.
Conversely,
GDM
[3] are
very demanding in CPU and memory resources
and are quite
diﬀicult to implement
. On the other hand, they are
well suited to take in account
complex geometries
as they might be run on unstructured meshes. Besides, their
formulation allows
local order reﬁnement
. Both these methods have been successfully
implemented in dedicated solvers, over the years. Based on this, the idea of a
heterogeneous method coupling
in order to split the computational domain and/or
locally take advantage of each method’s qualities has already been advanced in [2].
Motivations
The idea of such a coupling is then to be able to
take into account
complex boundary geometries
and boundary conditions running a DGM
on a
fully unstructured mesh around the obstacles and a
much cheaper
FDM on a cartesian grid further
away
.
Moreover, a coupled solver approach allows a
fully
parallel design
, so that DG and FD computations
are run on diﬀerent CPUs.
(i) Compared performances of DGM and FDM, both on a cartesian mesh
Here, we present preliminary results in a ﬁrst
attempt in the development and the study of a
coupling algorithm between DG solver
SPACE
[4]
,
and a dedicated FD solver. Our modelling is
presently based on 2D
Linearized Euler Equations
(LEE) and rigidwall boundary conditions.
We also introduce elements of the foreseen
developments, and the evaluation of the coupling
method.
We approximate solutions of 2D Linearized Euler Equations (LEE) :
within
+ imposed conditions on
: sound velocity
: acoustic perturbation velocity
: mean flow velocity
: acoustic perturbation density
: mean flow density
Implemented methods
Finite diﬀerences method
:
Discontinuous Galerkin method
(SPACE) :
‣
Centered stencils
of order up to
6
for gradients
approximation.
‣
Linear stabilizing spatialhighfrequencies
ﬁlter
: centered
11 points (10th order)
stencil.
‣
Boundary conditions are imposed using
ghost points
.
Note that the number of necessary ghost points is deﬁned by
the halfsize of the largest centered stencil.
‣
Lagrangian basis of order up to
8
are implemented.
‣
We use
fully upwind numerical ﬂuxes
at elements
boundaries.
Time integration method
:
‣
In both solvers, we use a 3step /
3rd order
Runge Kutta
scheme.
a)
Spatial approach
b)
Temporal approach
The coupling algorithm is based on a
minimal
overlapping
of the FD grid and the DG unstructured
mesh. Solvers do exchange informations at their
boundaries. As a ﬁrst approach, we impose two
requirements on meshes’ relative position.
‣
All ghost points of the FD grid lie within the DG
domain.
‣
All Gauss points at the DG couplingboundary lie
within the FD domain.
It allows :
‣
To
feed ghost points
of the
FD grid
with
interpolated values
of the solution ﬁeld using the
intrinsic polynomial basis used by the
DGM
.
‣
To
feed Gauss points
of the
DG coupling
boundary
using
interpolated values
of the solution
ﬁeld
over the FD grid
. In doing so, we compute and
impose a numerical ﬂux
to the boundary of the DG
domain.
For preliminary simplicity reasons, we impose both methods to use a common timestep and a common RK scheme.
Solvers do
exchange values at each RK subiteration
.
FIG. ii1 : A typical mesh overlapping
conﬁguration in this approach. FD grid in red, DG
mesh in blue. The FD domain boundary is notiﬁed
in black, note 5 rows of ghost points on its left.
In the following 2D test cases, we run
‣
a
DG P1
computation,
‣
a
6thorder
FD
gradient approximation stencils and a
10thorder
ﬁlter
.
In addition, note that
‣
Q1 interpolations
only are performed over the FD values.
‣
Acoustic perturbations are advected in a
ﬂuid at rest
.
a)
Plane wave advection
FIG. iii1 : Longitudinal cut. A sine wave is advected
form the DG domain (in blue) to the FD domain (in
red). Note the slight nonphysical oscillations in the
case of a coarser FD grid.
!
x
FD
=
L/
400
!
x
FD
=
L/
50
b)
Scattering of an acoustic mode by a circular obstacle
FIG. iii2 : A comparison
between a coupled and a
reference fullDG computation.
The domain is meshed as
displayed on top. FD coupling
points are not shown.
c)
Acoustic pulse at t=0
FIG. iii3 :
Propagation
of an acoustic
pulse from the
FD domain to
t h e D G
domain and
reﬂexion on a
c i r c u l a r
obstacle. The
d o ma i n i s
me s h e d a s
presented in
FIG. iii2.
FIG. iii4 : On
the same test
case, only the
FD domain is
p l o t t e d.
S p u r i o u s
reﬂexions from
the DG domain
are emphasized
by modifying
the color scale.
FIG. i1 : Linear advection of a sine wave.
FDM
DGM
FIG. i2 : L2 error VS computational time.
Note a gap in favour of FDM of a factor 4 to 7
in time for a given error.
Comparing DGM (P2 basis) and FDM (4th order + 10th order ﬁlter)
complemented by a common RK scheme leads to the following observations :
‣
DGM suﬀers a low optimal CFL (about 6 times lower than in the case of
FDM).
‣
Centered FDM schemes complemented by a high order ﬁlter oﬀer a good
control over numerical diﬀusion.
‣
In 2D cases, DGM requires about 2,6 times more single operations then
FDM for a single gradient approximation.
‣
This leads to the the result presented in
FIG i2
.
‣
Highorder interpolation over the FD grid.
In the presented examples, we only perform
Q1
interpolation
of the FD values in order to
reconstruct the solution ﬁeld at the DG domain’s
boundary. This, of course,
prevents from
preservi ng the method’s spati al order
.
Interpolations of a higher order (with respect to the
solver’s orders) are therefore required (see
FIG.
iv1
).
Although the ﬁrst attempts presented before show encouraging results, some further developments and
analysis are necessary to ensure the eﬀiciency of the method.
‣
Numerical testing
First, a numerical study of the
convergence rates of
the error
(in cases where an analytical solution is
available) of the twodomain coupled solver is
needed to make sure the
highorder is globally
maintained
. Secondly,
stability
in the case of
long
time
computations has to be tested.
FIG. iv1 : Overlapping DG and FD meshes. In
blue : FD coupling points. The boundary of the
DG domain (not plotted but lying on the left hand
side) is plotted in red. In orange, we select a
Gauss point and identify a quad of the FD grid
over which we perform a high order interpolation.
‣
Timeintegration approach
Imposing
common timesteps
and RK schemes in
both solvers is a
strong restriction
. In particular, it
may lead to a very low actual CFL in case of slightly
heterogeneous cell sizes. This issue can be solved by
performing
interpolations in time
, and adapting the
coupling algorithm.
‣
Applications and meshingstrategy
In the foreseen applications, meshes conﬁgurations
as presented before will have to be adapted. The idea
is to switch to a
Chimera approach
, with a
rectangular (
parallelepipedic)
DG domain (meant to
contain an obstacle) fully overlapped by a FD grid
(see.
FIG. iv2).
The coupling algorithm between
the twodomain case is then transposed to each
coupling boundary, at the faces of the DG
domain.
Rigid wall
Co u p l i n g
boundary to
DG domain
FIG. iv2 : Overlapping DG and FD meshes, in a
Chimera approach. In red : boundary of the DG
domain. In blue : FD coupling points. In white :
Chimera points whose values are initialized to 0
and never computed.
These results are encouraging. They show a good behaviour of the
coupled computations, in that particularly simple cases.
Computations are stable (at least at these ranges of time). Note that
non physical oscillations can be explained by both low interpolation
and DGM order.
Status
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