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Study of a high-order 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.

Dispersion-relation-preserving 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.

Runge-kutta discontinuous galerkin methods for convection-dominated 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 rigid-wall 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 spatial-high-frequencies

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 half-size 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 3-step /

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 coupling-boundary 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 sub-iteration

.

FIG. ii-1 : 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

6th-order

FD

gradient approximation stencils and a

10th-order

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. iii-1 : Longitudinal cut. A sine wave is advected

form the DG domain (in blue) to the FD domain (in

red). Note the slight non-physical 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. iii-2 : A comparison

between a coupled and a

reference full-DG computation.

The domain is meshed as

displayed on top. FD coupling

points are not shown.

c)

Acoustic pulse at t=0

FIG. iii-3 :

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. iii-2.

FIG. iii-4 : 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. i-1 : Linear advection of a sine wave.

FDM

DGM

FIG. i-2 : 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 i-2

.

‣

High-order 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.

iv-1

).

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 two-domain coupled solver is

needed to make sure the

high-order is globally

maintained

. Secondly,

stability

in the case of

long-

time

computations has to be tested.

FIG. iv-1 : 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.

‣

Time-integration 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 meshing-strategy

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. iv-2).

The coupling algorithm between

the two-domain 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. iv-2 : 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.

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