abstracts book!

Transcription

abstracts book!

Contents. . .
•
•
•
•
•
•
•
Welcome, Bem-vindo!
SHARK-FV week
Temptative schedule p.2
Hotel and restaurant p.2
Participants p.2
Abstracts p.5-34
Organisation p.36
Need for help?
Just ask!
•
•
•
•
•
•
Carolina Ribeiro
Stéphane Clain
Gaspar Machado
Jorge Figueiredo
Rui Pereira
Raphaël Loubère
[email protected]
Make sure you can leave!
Please take 1 minute during your
stay to make sure that you are
booked for a journey on a shuttle
back to the airport to take your
plane. Check with the locals.
Hotel contact information:
Hotel Parque do Rio,
Caminho Padre Manuel de Sá
Pereira, 4741-908 Fão, Portugal.
+351 253 981 521
GPS: 8 47.097 W, 41 31.192 N
Welcome, Bem-vindo!
Welcome to the first experimental Sharing Higher-order Advanced Know-how
on Finite Volume conference, SHARK-FV 2014 from April the 28th to May the
2nd.
The main purpose of this conference is to strengthen the collaborations between Finite Volume (FV) field actors and to share the burden of research and
development of numerical codes. This workshop brings the opportunity for
Researchers from International Universities and National Laboratories to discuss the State-of-the-Art of high(er)-order Finite Volume methods for a large
range of Physics and Engineering problems. The purposes of the workshops
are threefold
• to reinforce already existing collaborations;
• to create new interactions between researchers in the same field;
• to share detailed and technical experiences on specific issues and exchange ideas, numerical codes, test cases... any useful material.
The area of Esposende between Atlantic ocean and
Cavado river, the Hotel surrounded by luxurious private calm gardens and peaceful pine-woods, the famous Portuguese cuisine, provide a perfect incubator
for Exceptional science!
SHARK-FV week
The organisation of the workshop is as follows: Morning presentations are proposed followed by afternoon
sessions which are dedicated to intensive parallel workshops involving few collaborators.
We expect a lot of time to be dedicated to discussions and
small group working sessions, hence the few number of
participants!
Temptative schedule
Monday
8h00-9h00
9h00-9h50
9h50-10h40
10h40-11h00
11h00-11h50
11h50-12h40
12h40-13h20
13h30-14h30
14h30-16h30
16h30-19h30
20h00-
Tuesday
T ORO p.10
D IOT p.12
Opening (11h-)
L OUBÈRE p.5
F IGUEIREDO p.7
D UMBSER p.8
C OSTA p.14
M ACHADO p.17
Low Mach and
Kinetic HO meth.
Work
session
Dinner
Elliptic
and FVs
Work
session
Wednesday
Thursday
Friday
Breakfast
D IMARCO p.18
B LACHÈRE p.26 Closing
N ARSKI p.21
C LAIN p.29
—
Coffee break
R ISPOLI p.23
G ALLOUET p.31
—
V IGNAL p.24
V ÁSQUEZ p.33
—
Lunch
Lunch
Very high-order
Multi-physics,
—
FV methods
source terms
—
Work
Work
—
session
session
—
BANQUET
Dinner
Talks
Talks
◦ table
Free
The talks are scheduled to last for 45 minutes with 5 minutes of questions/answers. Abstracts are proposed in pages
5-34. After lunch, thematic round tables are organized for interested people, the goal being to exchange and share
news, raise new problematic and create a positive alchemy between the participants. Ideally such alchemy must
generate bombastic discussions to feed the following work sessions, or free time scheduled after 16h30.
This free time is left free for each participant to construct his ideal and optimal schedule for the week. Be careful that
some people may leave on Friday morning, so plan in advance your discussions!
Round tables will feature the following topics:
Elliptic
finite
volume
methods
In the context of elliptic
and parabolic systems of
equations one wonders how
do prospective high-order
finite volume methods
behave? Discussions may
revolve around this topic
and such approach.
Very high-order finite volumes methods
The discussion may concern
the needs of very high-order
schemes according to the
considered physics. It may
also deal with the techniques to be derived in order
to suitably improve existing
approaches.
Low Mach number and kinetic high-order methods
Sharing experiences with
new
techniques
and
schemes for all-speed flows
and kinetic equations will
be the focus of this dicussion as well as prospective
approaches (HPC, efficiency,
high-order, etc.).
Finite volume for multiphysics system and source
terms
Opportunity to discuss the
improvements and the relevance in deriving numerical schemes to simulate very
sophisticated models issuing
from strongly complex multifluid physics.
Hotel and restaurant
Participants
Hotel Parque do Rio (Caminho Padre Manuel de Sá Pereira) is located
40km North of Oporto, between the Atlantic Ofir Beach and the Cavado
River. Surrounded by luxurious private gardens and pinewoods offering
unforgettable peace and calm.
Florian Blachère, univ. Nantes
Stephane Clain, univ. do Minho
Ricardo Costa, univ. do Minho
Giacomo Dimarco, univ. Ferrara
Steven Diot, Los Alamos National
Laboratory
Mickael Dumbser, univ. Trento
Jorge Figueiredo, univ. do Minho
Francoise Foucher, univ. Nantes
Thierry Gallouët, univ. Marseille
Raphael Loubère, univ. Toulouse
Gaspar Machado, univ. do Minho
Jacek Narski, univ. Toulouse
Rui Pereira, univ. do Minho
Carolina Ribeiro, univ. do Minho
Vitorio Rispoli, univ. Toulouse
Khaled Saleh, univ. Marseille
Eleuterio Toro, univ. Trento
Elena Vásquez Cendón, univ. Santiago de Compostela
M-H Vignal, univ. Toulouse
• Several multi-purpose salons and restaurant. Large garden and
solarium, two swimming pools, snack-bar with outdoor seating.
• “Papa amoras” restaurant-bar and chef Rui Paula the acclaimed chef
of the DOC and DOP douro region restaurants, welcome us for
breakfasts at 7h-9h (7AM-9AM), lunches at 13h (1PM) and dinners
20h (8PM).
• Wednesday evening is scheduled the exceptional banquet at 20h
(8PM), tick your bookmark!
Back to Contents
2
List of abstracts
The next pages gather the abstracts of the talks given during SHARK-FV, here is a summary:
R. Loubère, M. Dumbser, S. Diot, page 5
COUPLING THE MULTI-DIMENSIONAL OPTIMAL ORDER DETECTION (MOOD) METHOD AND THE ARBITRARY
HIGH ORDER DERIVATIVES (ADER) APPROACHES
J. Figueiredo , S. Clain, C. Ribeiro, page 7
A NON-CONSERVATIVE FLUX APPROACH ENSURING THE C-PROPERTY FOR THE LAKE AT REST IN THE FRAMEWORK OF A MOOD BASED 2D SHALLOW-WATER SIXTH-ORDER FINITE VOLUME SCHEME
M. Dumbser, page 8
HIGH ORDER ONE-STEP AMR AND ALE FINITE VOLUME METHODS FOR HYPERBOLIC PDE
E. Toro, page 10
RECENT DEVELOPMENTS ON HIGH-ORDER ADER FINITE VOLUME SCHEMES
S. Diot, M.M. Francois, E.D. Dendy, page 12
A VERY-HIGH-ORDER SHARP INTERFACE METHOD TO SIMULATE MULTI-MATERIAL FLOWS BASED ON THE
MOOD CONCEPTS
R. Costa, S. Clain, G. J. Machado, page 14
FINITE VOLUME SCHEME BASED ON CELL-VERTEX RECONSTRUCTIONS FOR ANISOTROPIC DIFFUSION PROBLEMS WITH DISCONTINUOUS COEFFICIENTS
S. Clain, G.J. Machado, page 17
A HIGH-ORDER FINITE VOLUME SCHEME FOR THE TIME-DEPENDENT CONVECTION-DIFFUSION EQUATION
G. Dimarco, L. Pareschi, page 18
ASYMPTOTIC PRESERVING IMPLICIT-EXPLICIT RUNGE-KUTTA METHODS FOR NON LINEAR KINETIC EQUATIONS
G. Dimarco, J. Narski, R. Loubère, page 21
TOWARDS AN ULTRA EFFICIENT KINETIC SCHEME : HIGH-PERFORMANCE COMPUTING
G. Dimarco, R. Loubère, V. Rispoli, page 23
A MULTI-SCALE FAST SEMI-LAGRANGIAN METHOD FOR RAREFIED GAS DYNAMICS
G. Dimarco, R. Loubère, M.-H. Vignal, page 24
FINITE VOLUMES SCHEMES PRESERVING THE LOW MACH LIMIT FOR EULER SYSTEMS
F. Blachère, R.Turpault, page 26
AN ASYMPTOTIC AND ADMISSIBILITY PRESERVING FINITE VOLUME SCHEME FOR SYSTEMS OF CONSERVATION
LAWS WITH SOURCE TERMS ON 2D UNSTRUCTURED MESHES
S. Clain, G.J. Machado, R.M.S. Pereira, A. Boularas, page 29
VERY HIGH ORDER FINITE VOLUME SCHEME FOR THE 2D AND 3D LINEAR CONVECTION DIFFUSION EQUATION
T. Gallouet, page 31
COMPACTNESS OF SEQUENCES OF APPROXIMATE SOLUTIONS OF PARABOLIC EQUATIONS
A. Bermúdez, S. Busto, M. Cobas, J.L. Ferrín, L. Saavedra, M.E. Vázquez-Cendón, page 33
A HIGH ORDER FINITE VOLUME/FINITE ELEMENT PROJECTION METHOD FOR LOW-MACH NUMBER FLOWS
WITH TRANSPORT OF SPECIES
3
SHARK-FV — April 28 - May 2 2014 — Ofir, Portugal.
Sharing Higher-order Advanced Research Know-how on Finite Volumes
COUPLING THE MULTI-DIMENSIONAL OPTIMAL ORDER DETECTION
(MOOD) METHOD AND THE ARBITRARY HIGH ORDER DERIVATIVES
(ADER) APPROACHES
R. Loubère b∗ , M. Dumbser a , S. Diot c
a
Laboratory of Applied Mathematics. Department of Civil, Environmental and Mechanical Engineering, University of
Trento, Via Mesiano 77, I-38123 Trento (TN), Italy.
b Institut de Mathématiques de Toulouse, Université de Toulouse 31062 Toulouse, France.
c Los Alamos National Laboratory, Los Alamos, NM, 87545
ABSTRACT
In this paper, we investigate the coupling of the Multi-dimensional Optimal Order Detection (MOOD)
method and the Arbitrary high order DERivatives (ADER) approach in order to design a new high order
accurate, robust and computationally efficient Finite Volume (FV) scheme dedicated to solve nonlinear
systems of hyperbolic conservation laws on unstructured triangular and tetrahedral meshes in two and
three space dimensions, respectively. The Multi-dimensional Optimal Order Detection (MOOD) method
for 2D and 3D geometries has been introduced in a recent series of papers [1, 2, 3] for mixed unstructured meshes. It is an arbitrary high-order accurate Finite Volume scheme in space, using polynomial
reconstructions with a posteriori detection and polynomial degree decrementing processes to deal with
shock waves and other discontinuities. In the following work, the time discretization is performed with
an elegant and efficient one-step ADER procedure [4, 5]. Doing so, we retain the good properties of the
MOOD scheme, that is to say the optimal high-order of accuracy is reached on smooth solutions, while
spurious oscillations near singularities are prevented. The ADER technique permits not only to reduce
the cost of the overall scheme as shown on a set of numerical tests in 2D and 3D, but it also increases the
stability of the overall scheme. A systematic comparison between classical unstructured ADER-WENO
schemes and the new ADER-MOOD approach has been carried out for high-order schemes in space and
time in terms of cost, robustness, accuracy and efficiency. The main finding of this paper is that the combination of ADER with MOOD generally outperforms the one of ADER and WENO either because at
given accuracy MOOD is less expensive (memory and/or CPU time), or because it is more accurate for a
given grid resolution. A large suite of classical numerical test problems has been solved on unstructured
meshes for three challenging multi-dimensional systems of conservation laws: the Euler equations of
compressible gas dynamics, the classical equations of ideal magneto-Hydrodynamics (MHD) and finally
the relativistic MHD equations (RMHD), which constitutes a particularly challenging nonlinear system
of hyperbolic partial differential equation. All tests are run on genuinely unstructured grids composed of
simplex elements.
ACKNOWLEDGMENTS
M.D. has been financed by the European Research Council (ERC) under the European Union’s Seventh
Framework Programme (FP7/2007-2013) with the research project STiMulUs, ERC Grant agreement no. 278267.
R.L. has been partially funded by the ANR under the JCJC project “ALE INC(ubator) 3D”. This work has
been authorized for publication under the reference LA-UR-13-28795. The authors would like to acknowledge PRACE for awarding access to the SuperMUC supercomputer based in Munich, Germany at the Leibniz
Rechenzentrum (LRZ).
∗ Correspondence
to [email protected]
5
REFERENCES
[1] S. Clain, S. Diot, and R. Loubère. A high-order finite volume method for systems of conservation lawsmulti-dimensional optimal order detection (MOOD). Journal of Computational Physics, 230(10):4028 –
4050, 2011.
[2] S. Diot, S. Clain, and R. Loubère. Improved detection criteria for the multi-dimensional optimal order
detection (MOOD) on unstructured meshes with very high-order polynomials. Computers and Fluids,
64:43 – 63, 2012.
[3] S. Diot, R. Loubère, and S. Clain. The MOOD method in the three-dimensional case: Very-high-order
finite volume method for hyperbolic systems. International Journal of Numerical Methods in Fluids,
73:362–392, 2013.
[4] M. Dumbser. Arbitrary high order PNPM schemes on unstructured meshes for the compressible Navier–
Stokes equations. Computers & Fluids, 39:60–76, 2010.
[5] M. Dumbser, M. Castro, C. Parés, and E.F. Toro. ADER schemes on unstructured meshes for nonconservative hyperbolic systems: Applications to geophysical flows. Computers and Fluids, 38:1731 –
1748, 2009.
6
A NON-CONSERVATIVE FLUX APPROACH ENSURING THE C-PROPERTY
FOR THE LAKE AT REST IN THE FRAMEWORK OF A MOOD BASED 2D
SHALLOW-WATER SIXTH-ORDER FINITE VOLUME SCHEME
J. Figueiredo a∗ , S. Clain a , C. Ribeiro a
a
Departamento de Matemática e Aplicações and Centro de Matemática, Universidade do Minho, Campus de Gualtar 4710-057 Braga, Portugal.
ABSTRACT
We solve the 2D shallow-water problem with bathymetric source term using a finite volume scheme
that combines a Polynomial Reconstruction Operator (PRO) with a Multi-dimensional Optimal Detection
(MOOD). This numerical scheme allows to obtain high-order of accuracy (up to sixth-order) for smooth
solutions. This is the first use of the PRO-MOOD technique in a non-conservative problem. Still, a
relatively basic property such as the C-property for the lake at rest is not guaranteed when there are
bathymetry variations, unless a specific approach is used. This is a very important issue when simulating,
for instance, Tsunami wave propagation with varying bathymetry, since one wants to avoid the appearance
and development of small oscillations in the sea regions as long as they remain unperturbed by true waves.
We show that the lake at rest can be reproduced exactly using a non-conservative flux together with the
classic conservative one independently of the overall scheme order. Furthermore, we show that this flux
scheme can be applied as long as the conservative flux being used admits a so-called physical bathymetry
representative (a new concept introduced specifically for this purpose), which is the case, for instance, of
the Rusanov, HLL, and HLLC fluxes, even under relatively stiff bathymetries.
∗ Correspondence
7
8
HIGH ORDER ONE-STEP AMR AND ALE FINITE VOLUME METHODS
FOR HYPERBOLIC PDE
M. Dumbser b∗ ,
a
Laboratory of Applied Mathematics. Department of Civil, Environmental and Mechanical Engineering, University of
Trento, Via Mesiano 77, I-38123 Trento (TN), Italy.
ABSTRACT
In this talk we present a unified family of high order accurate finite volume and discontinuous Galerkin
finite element schemes on moving unstructured and adaptive Cartesian meshes for the solution of conservative and non-conservative hyperbolic partial differential equations.
The PN PM approach adopted here uses piecewise polynomials uh of degree N to represent the data in
each cell. For the computation of fluxes and source terms, another set of piecewise polynomials wh of
degree M ≥ N is used, which is computed from the underlying polynomials uh using a reconstruction
or recovery operator. The PNPM method contains classical high order finite volume schemes (N = 0)
and high order discontinuous Galerkin (DG) finite element methods (N = M) as two special cases of a
more general class of numerical schemes. The schemes are derived in general ALE form so that Eulerian
schemes on fixed meshes and Lagrangian schemes on moving meshes can be recovered as special cases of
the ALE formulation. Furthermore, the method can also be naturally implemented on space-time adaptive
Cartesian grids (AMR), together with time-accurate local time stepping (LTS). To assure the robustness
of the method at discontinuities, a nonlinear WENO reconstruction is performed. The time integration is
carried out in one single step using a high order accurate local space-time Galerkin predictor that is also
able to deal with stiff source terms.
Applications are shown for the compressible Euler and Navier-Stokes equations, for the MHD equations
and for the Baer-Nunziato model of compressible multi-phase flows.
REFERENCES
[1] M. Dumbser, A. Uriuuntsetseg, O. Zanotti. On ALE-Type One-Step WENO Finite Volume Schemes for
Stiff Hyperbolic Balance Laws. Communications in Computational Physics, 14:301–327, 2013.
[2] W. Boscheri and M. Dumbser. Arbitrary-Lagrangian-Eulerian One-Step WENO Finite Volume Schemes
on Unstructured Triangular Meshes. Communications in Computational Physics, 14:1174–1206, 2013.
[3] M. Dumbser and W. Boscheri. High-Order Unstructured Lagrangian One-Step WENO Finite Volume
Schemes for Non-Conservative Hyperbolic Systems: Applications to Compressible Multi-Phase Flows.
Computers and Fluids, 86:405–432, 2013.
[4] M. Dumbser, O. Zanotti, A. Hidalgo and D.S. Balsara. ADER-WENO Finite Volume Schemes with SpaceTime Adaptive Mesh Refinement. Journal of Computational Physics, 248:257–286, 2013.
[5] M. Dumbser, A. Hidalgo and O. Zanotti. High Order Space-Time Adaptive ADER–WENO Finite Volume Schemes for Non-Conservative Hyperbolic Systems. Computer Methods in Applied Mechanics and
Engineering, 268:359–387, 2014.
∗ Correspondence
9
10
RECENT DEVELOPMENTS ON HIGH-ORDER ADER FINITE VOLUME
SCHEMES
Eleuterio Toro a
a
University of Trento, Trento , Italy
ABSTRACT
First I briefly review the ADER approach for solving hyperbolic balance laws to arbitrary order of accuracy in both space and time. Then I present a new, locally implicit solver for the Generalized Riemann
Problem, the building block of ADER methods. The resulting explicit ADER schemes are then capable
of dealing with balance laws with stiff source terms. Finally I describe the application of the new version
of ADER to solve general advection-diffusion reaction equations, that may also include source terms,
reformulated as hyperbolic systems of balance laws via a relaxation procedure. Numerical examples are
shown.
11
12
A VERY-HIGH-ORDER SHARP INTERFACE METHOD TO SIMULATE
MULTI-MATERIAL FLOWS BASED ON THE MOOD CONCEPTS
S. Diot a∗ , M.M. François a , E.D. Dendy b
a
b
Fluid Dynamics and Solid Mechanics (T-3), Los Alamos National Laboratory, NM 87545, USA.
Computational Physics and Methods (CCS-2), Los Alamos National Laboratory, NM 87545, USA.
ABSTRACT
Simulations of multi-material compressible flows are of crucial interest for industrial and fundamental
researches as many physical complex phenomena are driven by such hydrodynamics. Let us cite the
explosion of a star, transport in gas carrier or Inertial Confinement Fusion (ICF) for instance. The development of more robust and efficient methods for these complex problems is challenging but necessary to
shorten the computational time to get the solution and therefore improve the current prediction capabilities. Aiming at this objective, I will present an original direct Eulerian Volume-Of-Fluid (VOF) method on
unstructured meshes that uses ideas that are analogous to the ones in [3] in order to get a better approximation of the volumes that are fluxed through the faces. This scheme better approximates the compressibility
of the velocity field and therefore allows to limit the mass exchange occurring at the material interface.
The approach is validated for a purely advective problem of several materials on unstructured meshes.
Comparisons to exact solutions will show that the approach reduces the errors, in particular when the flow
is incompressible. Then we will apply this method to a reduced model for compressible material-fluid
flows with single velocity and instantaneous pressure equilibrium. We will emphasize the advantages
of this method and point out the gain in accuracy obtained on classical test cases. In addition, we will
study the efficiency of the proposed method by monitoring the ratio between errors and computational
times. Finally, preliminary results of the coupling between this method and the recently developed Multidimensional Optimal Order Detection (MOOD) method [1, 2] will be provided. This shall demonstrate
that designing a higher-order method through the MOOD concepts allows to reach optimal order while
ensuring the method robustness.
ACKNOWLEDGMENT
This work was performed under the auspices of the National Nuclear Security Administration of the US
Department of Energy at Los Alamos National Laboratory under Contract No. DE-AC52-06NA25396 and
supported by the DOE Advanced Simulation and Computing (ASC) program. Approved for public release
under LA-UR-13-29379.
REFERENCES
[1] S. Diot, S. Clain, R. Loubère, Improved detection criteria for the Multi-dimensional Optimal Order Detection (MOOD) on unstructured meshes with very high-order polynomials, Comput. Fluids 64 (2012)
43–63.
[2] S. Diot, R. Loubère, S. Clain, The Multidimensional Optimal Order Detection method in the threedimensional case: very high-order finite volume method for hyperbolic systems, Int. J. Numer. Meth.
Fluids 73 (2013) 362–392.
[3] J. López, H. Hernández, P. Gómez, F. Faura, A volume of fluid method based on multidimensional advection and spline interface reconstruction, J. Comp. Phys. 195 (2004) 718–742.
∗ Correspondence
13
14
FINITE VOLUME SCHEME BASED ON CELL-VERTEX
RECONSTRUCTIONS FOR ANISOTROPIC DIFFUSION PROBLEMS WITH
DISCONTINUOUS COEFFICIENTS
R. Costa a∗ , S. Clain a,b , G. J. Machado a
a
b
Departamento de Matemática e Aplicações e Centro de Matemática, Campus de Gualtar - 4710-057 Braga, Portugal.
Institut de Mathématiques de Toulouse, Université de Toulouse 31062 Toulouse, France.
ABSTRACT
We propose a new second-order finite volume scheme for non-homogeneous and anisotropic diffusion
problems based on cell to vertex reconstructions involving minimization of functionals to provide the
coefficients of the cell to vertex mapping. The method handles complex situations such as large preconditioning number diffusion matrices and very distorted meshes. Numerical examples are provided to show
the effectiveness of the method.
INTRODUCTION
The design of second-order schemes is still a challenging and important question. In fact, very high-order
methods are rather complicated and require an important implementation effort whereas second-order methods are quite simple and easy to code. A popular class of second-order finite volume schemes is based on
vertex reconstructions using point-wise approximations on cells associated to a specific point location (usually
the centroid). Then combining cell and vertex values, gradient approximations are evaluated to compute the
diffusive flux on the interfaces.
In [2] we consider the homogeneous and isotropic situation where we evaluate the vertex values ψn from
the cell values φi based on a simple linear combination to compute the coefficients βni . In the present study we
extend the previous method to non-homogeneous and anisotropic problems.
FORMULATION
Let Ω be an open bounded polygonal domain of R2 with boundary ∂ Ω. We split Ω into two non-overlapping
subdomains Ω1 and Ω2 sharing a common interface Γ where the diffusion tensor K is discontinuous with K = K1
in Ω1 and K = K2 in Ω2 . We seek function φ = φ1 in Ω1 and φ = φ2 in Ω2 , solution of the anisotropic steadystate diffusion equations
∇ · (−K1 ∇φ1 ) = f1 ,
in Ω1 ,
∇ · (−K2 ∇φ2 ) = f2 ,
in Ω2 ,
where the source terms f1 and f2 are regular functions on Ω1 and Ω2 , respectively. We prescribe the continuity
both for the flux and the function, namely K1 ∇φ1 · n = K2 ∇φ2 · n and φ1 = φ2 on Γ, the Dirichlet condition on
∂ Ω = ΓD and the Neumann condition on ∂ Ω = ΓN .
CELL-VERTEX MAPPING
The generic finite volume discretization for each cell ci , i = 1, . . . , I cast in the residual form writes
Gi =
∗ Correspondence
|ei j |
Fi j − fi ,
|ci |
j∈ν(i)
∑
15
where Fi j is an approximation of the diffusive flux through the edge ei j and fi is an approximation of the mean
value of f over the cell ci .
Let φi , i = 1, . . . , I, be an approximation of φ on the mass centre qi of cell ci , and let ψn , n = 1, . . . , N, be
an approximation of φ on vertex vn . The goal is the design of a procedure to compute ψn from the neighbour
point-wise cell values. We define ψn as
ψn = ∑ βni φi ,
i∈µ(n)
where µ(n) is the set of the indices of the neighbour cells. Let gather in vector Bn = (βni )i∈µ(n) the coefficients
of the linear combination of the cell data. We seek Bn such that Λ1 (Bn ) = 1, Λ2 (Bn ) = 0, and Λ3 (Bn ) = 0 where
Λ1 (Bn ) =
∑
Λ2 (Bn ) =
βni ,
i∈µ(n)
∑
i∈µ(n)
∑
Λ3 (Bn ) =
βni (qix − vnx ),
i∈µ(n)
βni (qiy − vny ),
in order to preserver first degree polynomials. To deal with situations where the solution is not unique, which
happens when there are more than three cells in µ(n), we introduce the quadratic functional
E(Bn ) =
1
∑ ωni (βni − θni )2 ,
2 i∈µ(n)
where θni are the target values such that ∑i∈µ(n) θni = 1, and ωni are positive weights. We then solve the problem
with the classical minimization with Lagrange multipliers where Λ1 , Λ2 and Λ3 are the constraints.
At last, we compute the flux approximation Fi j based on a polynomial which interpolates the vertex values
of ei j and the cell values of ci and c j . We compute the source term approximation fi based on the cell value φi
and on the vertex values of ψn of each vn ∈ ci .
NUMERICAL RESULTS
We briefly present a simulation where we consider the domain Ω = ]0, 1[2 and the discontinuous diffusion
tensor with
1 0
100
0
K1 =
,
K2 =
.
0 1
0 0.01
The source terms are given by f1 (x, y) = 2π 2 cos(πx) sin(πy) on Ω1 =]0, 0.5[×]0, 1[ and by f2 (x, y) = (1 +
0.012 ) cos(πx) sin(πy) on Ω2 =]0.5, 1[×]0, 1[. We compute the numerical solution using a triangular Delaunay
mesh (see an example of the mesh in Fig.1, left) and we represent the result in Fig. 1 (right).
y
y
1
1
0.75
0.75
0.5
0.5
0.25
0.25
x
0
0
0.25
0.5
0.75
x
0
1
0
0.25
-0.01
0.5
0.495
0.75
1
1
FIGURE 1: Delaunay Mesh (left) and numerical solution on a fine mesh (right).
REFERENCES
[1] Clain, S., Machado, G. J., Nóbrega, J. M., Pereira, R. M. S.: A sixth-order finite volume method for the
convection-diffusion problem with discontinuous coefficients, Computer Methods in Applied Mechanics
and Engineering 267, 43–64 (2013).
[2] Costa, R., Clain, S., Machado, G. J.: New cell-vertex reconstruction for finite volume scheme: application
to the convection-diffusion-reaction equation, under review.
16
A HIGH-ORDER FINITE VOLUME SCHEME FOR THE TIME-DEPENDENT
CONVECTION-DIFFUSION EQUATION
S. Clain a,b , G.J. Machado a∗
a
b
Centre of Mathematics, University of Minho, Campus de Azurém, Guimarães, Portugal.
Institut de Mathématiques de Toulouse,Université de Toulouse, 31062 Toulouse, France.
ABSTRACT
The time discretization of a very high-order finite volume method may give rise to new numerical difficulties resulting into accuracy degradations. Indeed, for the simple one-dimensional time-dependent
convection-diffusion equation for instance, a conflicting situation between the source term time discretization and the boundary conditions may arise when using the standard Runge-Kutta method. We propose an
alternative procedure by extending the Butcher Tableau to overcome this specific difficulty and achieve
fourth-, sixth- or eighth-order of accuracy schemes in space and time. To this end, a new finite volume
method is designed based on specific polynomial reconstructions for the space discretization, while we
use the Extended Butcher Tableau to perform the time discretization. A large set of numerical tests has
been carried out to validate the proposed method (see [1] and [2]).
REFERENCES
[1] S. Clain, G.J. Machado, J.M. Nobrega, R.M.S. Pereira. A sixth-order finite volume method for multidomain
convection-diffusion problem with discontinuous coefficients. Computer Methods in Applied Mechanics
and Engineering, 267, 43–64, 2013.
[2] S. Clain, G.J. Machado. A very high-order finite volume method for the time-dependent convectiondiffusion problem with Butcher tableau extension (submitted).
∗ Correspondence
17
18
ASYMPTOTIC PRESERVING IMPLICIT-EXPLICIT RUNGE-KUTTA
METHODS FOR NON LINEAR KINETIC EQUATIONS.
G. Dimarco b∗ , L. Pareschi a,b
a
Department of Mathematics and Computer Science, University of Ferrara, Italy,
Department of Mathematics and Computer Science, University of Ferrara, Italy &
b
ABSTRACT
In this work, we discuss Implicit-Explicit (IMEX) Runge Kutta methods which are particularly adapted to
stiff kinetic equations of Boltzmann type. We consider both the case of easy invertible collision operators
and the challenging case of Boltzmann collision operators. We give sufficient conditions in order that
such methods are asymptotic preserving and asymptotically accurate. Their monotonicity properties are
also studied. In the case of the Boltzmann operator, the methods are based on the introduction of a
penalization technique for the collision integral. This reformulation of the collision operator permits to
construct penalized IMEX schemes which work uniformly for a wide range of relaxation times avoiding
the expensive implicit resolution of the collision operator. Finally we show some numerical results which
confirm the theoretical analysis.
INTRODUCTION
The numerical solution of Boltzmann-type equations close to fluid regimes represents a real challenge for
numerical methods. In these regimes, in fact, the intermolecular collision rate grows exponentially and the
collisional time becomes very small. On the other hand, the actual time scale for evolution is the fluid dynamic
time scale, which can be much larger than the collisional time. A non dimensional measure of the importance
of collision is given by the Knudsen number which is large in the rarefied regions and small in the fluid ones.
Standard computational approaches lose their efficiency due to the necessity of using very small time steps in
deterministic schemes or, equivalently, a large number of collisions in probabilistic approaches. Unfortunately
the use of implicit solvers originates a prohibitive computational cost due to the high dimensionality and the
nonlinearity of the collision operator.
In this talk, we develop Implicit-Explicit Runge-Kutta methods [1, 2] which are particularly efficient for stiff
non linear kinetic equations. First we consider the case where the implicit inversion of the collision term does
not represent a problem, like for example the case of simple BGK operators. Asymptotic preservation properties
and monotonicity are carefully studied and analyzed. Subsequently we deal with the challenging case of the
full Boltzmann equation. To this aim, we introduce a penalization strategy based on a decomposition of the
gain term of the collision operator into an equilibrium and a non equilibrium part. This permits to derive new
penalized IMEX schemes which keep the good asymptotic preservation properties of standard IMEX schemes
by avoiding the costly inversion of the collision term.
The methods studied are based on the following decomposition
R(Y ) = N(Y ) + L(Y ),
(1)
where N(Y ) represents the non-dissipative non-linear part and L(Y ) is a linear term such that L(Y ) = 0 implies
Y = E(y). For example L(Y ) = A(E(y) − Y ) where A > 0 is an estimate of the Jacobian of R evaluated at
∗ Correspondence
19
equilibrium. Note that, at variance with standard linearization techniques which operate on the short time scale,
the operator is linearized on the asymptotically large time scale. This decomposition permits to apply IMEX
techniques which are implicit in the linear part and explicit in the non-linear part. The use of such techniques,
as we will see, permits to achieve unconditionally stable and asymptotic preserving methods at the cost of an
explicit scheme.
REFERENCES
[1] G. Dimarco and L. Pareschi. Asymptotic preserving implicit-explicit Runge-Kutta methods for non linear
kinetic equations. SIAM Journal of Numerical Analysis, Vol. 51, pp. 1064-1087 (2013)
[2] L. Pareschi and G. Russo Implicit-Explicit Runge-Kutta methods and applications to hyperbolic systems
with relaxation. J. Sci. Comput., 25 (2005), 129–155.
20
TOWARDS AN ULTRA EFFICIENT KINETIC SCHEME :
HIGH-PERFORMANCE COMPUTING
G. Dimarco a , J. Narski b∗ , R. Loubère b
a
b
Department of Mathematics and Computer Science. University of Ferrara, Italy.
ABSTRACT
In this paper we present the new 3D/3D fast kinetic scheme developed in [1, 2] in its parallel version on
classical architecture using OPEN-MP and on GPU architecture using CUDA. The goal is to prove that
this new scheme is well adapted to any type of parallelisation, and that the gain in CPU time is substantial
on nowadays affordable computer. We briefly present the sequential version of our kinetic scheme and
focus on important details for a parallel implementation. Numerical tests are shown for the full 3D/3D
simulations. These assess the very interesting speed-up factor gain between the sequential code and the
parallel version.
INTRODUCTION
In many applications, the correct physical solution of systems far from thermodynamic equilibrium (rarified
gases, plasma) require resolution of kinetic equations. These simulations are typically very resource consuming
due to the large dimension of the problem. Indeed, the distribution function depends on seven independent
variables: three space coordinates, three coordinates in the velocity space and the time.
Dimarco and Loubère have recently proposed in [1, 2] a method based on a splitting technique called Fast
Kinetic Solver (FKS). The method relies on a Lagrangian transport scheme and a discretization of the velocity
space in the framework of so-called Discrete Velocity Models (DVM), where the velocity is discretized into a
set of fixed velocities. The original equation is replaced by a set of linear transport equations plus a coupling
interaction term corresponding to the collision operator. This allowed to drastically reduce the computational
cost of the transport part. The FKS scheme was shown to be capable of solving the full six dimensional problem
(3D in space + 3D in velocity) on a single laptop on reasonable meshes (1003 in space, 123 in velocity).
Although the simulations were possible, the computational time remained long. The purpose of this work is to
explore a possibility of parallelization of the FKS method in order to reduce the computational time on multi
core shared memory system and/or GPU.
PROBLEM FORMULATION
In this work, the following Boltzman-BGK equation is considered
1
V · ∇X f = (M f − f ),
∂t f +V
τ
(1)
X ,V
V ,t) of particles at position X ∈ Ω ⊂ Rdx , at time t > 0 and which move
describing the distribution f = f (X
d
v
with velocity V ∈ R . We consider the general case in which we have dx = dv = d = 3 dimensions in space
and velocity. The interaction operator is the BGK operator: collisions are modeled by a relaxation towards the
∗ Correspondence
21
FIGURE 1: Kelvin Helmholtz instability obtained with GPU simulations at time steps: 4000, 6000 and 8000.
local thermodynamical equilibrium defined by the Maxwellian distribution function M f
U −V
V k2
−kU
ρ
U , T ] (V
V) =
M f = M f [ρ,U
exp
,
2RT
(2πθ )d/2
(2)
where ρ ∈ R and U ∈ Rdv are the density and mean velocity, T is the temperature and R the gas constant.
The BGK equation is first discretized in the velocity space. The discrete velocity BGK model consists then
of a set of N evolution equations in the velocity space for fk of the form
1
V k · ∇X fk = (Ek [F] − fk ),
∂t fk +V
τ
(3)
where Ek [F] is a suitable approximation of M f and Nv is the total number of velocity grid points.
Next, the splitting technique is applied to every equation of (3). First the transport step is solved exactly,
providing initial data for the relaxation step:
V k · ∇X fk = 0,
∂t fk +V
1
Relaxation stage −→ ∂t fk = (Ek [F] − fk ).
τ
The equations a re finally discretized in space on a uniform Cartesian grid consisting of Ns points.
Transport stage −→
(4)
(5)
Parallelization
The main computational difficulty is related to the resolution of the relaxation stage, as it involves evaluation
of exponential functions. Fortunately, this stage can be easily decoupled into Ns ×Nv separate equations, each of
them solved separately in parallel. Indeed, (5) involves no interaction between fk and fi for k 6= i and the space
cells are independent. The density, velocity and temperature used for computation of Ek [F] can be precalculated
for every space cell prior to the relaxation stage. The parallelization can be done in a following way :
1. Transport of particles. Move in parallel Nv particles.
2. Relaxation step. Perform in parallel the relaxation step for Nv × Ns particles, parallelization is performed
on the external loop over Nv particles.
3. Update primitive variables. Update the density, mean velocity and temperature for every space cell.
In the numerical experiments performed this method showed a speed-up close to perfect on shared memory
systems with OMP interface. GPU parallelization allowed to obtain Kelvin Helmholtz instability on a 2003
space and 103 velocity grids in less then 32 hours — compared to 200 DAYS on a sequential machine.
REFERENCES
[1] G. D IMARCO , R. L OUBÈRE, Towards an ultra efficient kinetic scheme. Part I: basics on the BGK equation, J. Comput. Phys., Vol. 255, 2013, pp 680-698.
[2] G. D IMARCO , R. L OUBÈRE, Towards an ultra efficient kinetic scheme. Part II: the high order case, J.
Comput. Phys., Vol. 255, 2013, pp 699-719.
22
A MULTI-SCALE FAST SEMI-LAGRANGIAN METHOD FOR RAREFIED
GAS DYNAMICS
G. Dimarco a , R. Loubère b , V. Rispoli b∗
a
b
Department of Mathematics and Computer Science, University of Ferrara, 44100 Ferrara, Italy.
Institut de Mathématiques de Toulouse, Université de Toulouse, 31062 Toulouse, France.
ABSTRACT
In this talk we present an extension of the method developed in [1, 2] aiming at the numerical resolution
of multi-scale problems arising in rarefied gas dynamics. The scope of this work is to consider situations
in which the whole domain does not demand the use of a kinetic model everywhere. In many realistic
applications some regions of the domain require a microscopic description, given by kinetic equations,
while the rest of the domain can be treated with a coarser model of fluid type. Our aim is to show
that the kinetic scheme developed in previously cited articles is perfectly suited for building domain
decomposition strategies. Exploiting the latter, the numerical scheme’s efficiency greatly increases and
it becomes very attractive with respect to classical numerical techniques for kinetic equations and multiscale realistic problems. Several numerical results in the two and three dimensional settings are presented,
which were obtained saving computational resources (CPU, memory and time). The presented work is
the object of a recent in-review paper [3].
REFERENCES
[1] G. Dimarco and R. Loubère. Towards an ultra efficient kinetic scheme. Part I: basics on the BGK equation.
Journal of Computational Physics, vol. 255, pp. 680–698 (2013)
[2] G. Dimarco and R. Loubère. Towards an ultra efficient kinetic scheme. Part II: the high order case. Journal
of Computational Physics, Vol. 255, pp. 699-719 (2013)
[3] G. Dimarco, R. Loubère and V. Rispoli, A multiscale fast semi-Lagrangian method for rarefied gas dynamics. Submitted to Journal of Computational Physics.
∗ Correspondence
23
24
FINITE VOLUMES SCHEMES PRESERVING THE LOW MACH LIMIT FOR
EULER SYSTEMS
G. Dimarco a,b , R. Loubère c , M.-H. Vignal b
a
Mathematics Institute of Toulouse, University Toulouse 3, France.
Mathematics & Computer Science Department, University of Ferrara, Italy.
c Mathematics Institute of Toulouse, CNRS, France.
b
ABSTRACT
I am interested in the so-called Asymptotic preserving schemes. These schemes are well known to be
well adapted for the resolution of multiscale problems in which several regimes are present.
I will present the particular case of the low-Mach limit for Euler systems used in gas or fluid dynamics.
The square of the Mach number is the ratio of the kinetic and thermic energies of the fluid. When this
ratio tends to zero, the pressure waves are very fast and this yields the fluid incompressible.
When a standard explicit finite volume scheme is used, it is well known that its time step is constrained
by the C.F.L. (Courant-Friedrichs, Levy) condition. In the low-Mach regime, this leads to time steps
inversely proportional to the pressure waves velocity which is very large. Thus, explicit schemes suffer
from a severe numerical constraint in low-Mach regimes. Furthermore, these schemes are not consistent
in this regime. This means that they do not capture the incompressible limit.
Then, it is necessary to develop new schemes for bypassing these limitations. These new schemes must
be stable and consistent in all regimes: from low Mach numbers to order one Mach numbers
I will show how to construct such a scheme for Euler systems and I will present numerical results showing
the good behavior of these schemes in all regimes.
25
26
AN ASYMPTOTIC AND ADMISSIBILITY PRESERVING FINITE VOLUME
SCHEME FOR SYSTEMS OF CONSERVATION LAWS WITH SOURCE
TERMS ON 2D UNSTRUCTURED MESHES
F. Blachère a∗ , R. Turpault a
a Laboratoire
de Mathématiques Jean Leray 2, rue de la Houssinière - BP 92208 44322 Nantes, France.
ABSTRACT
We introduce a new finite volume technique to obtain numerical schemes which preserve both asymptotic and the set of admissible states on 2D unstructured meshes for the class of hyperbolic systems of
conservation laws with source terms described in [1].
The objective of this work is to design a suitable finite volume scheme to approximate the solutions of
hyperbolic systems of conservation laws with source terms which could be written as:
∂t U + div F(U) = −γ(U)R(U),
(1)
where U ∈ RN , F is a smooth function whose Jacobian has real eigenvalues, γ ≥ 0 and R is a smooth function
which fulfills the compatibility conditions stated in [1].
The main characteristic of such systems is to degenerate when γ t → ∞ into smaller parabolic systems of the
form:
∂t u − div M (u)∇u = 0,
(2)
where u ∈ Rn is related to U and M is a positive function or a symmetric, positive and definite matrix.
Examples of such systems include Euler equations with friction, the M1-model for radiative transfer, and
the telegraph equations.
From the numerical point of view, the main difficulty is to construct asymptotic preserving schemes, i.e.
schemes that degenerate into consistent schemes for the diffusion equation (2). This question has been a major
issue during the last decade and several efficient asymptotic preserving schemes, based on different techniques,
exist in 1D. In 2D, however, the problem is much more difficult. One of the reasons is that the target scheme in
the diffusion limit is often the classical two-point flux (a.k.a. FV4), which is not consistent anymore on general
meshes.
Up to now, only two examples of such suitable 2D schemes exist for non admissible meshes: the MPFAbased scheme developed in [2] and the one constructed from the diamond scheme in [4]. Nevertheless, in both
cases, it is not possible to ensure the preservation of the set of admissible states on general meshes. Obviously,
this property is critical in several configurations.
To overcome these difficulties, we propose a numerical procedure based on the technique developed by
Droniou and Le Potier in [3] to obtain positivity-preserving discretization of the gradient for parabolic or
elliptic equations. This technique serves as a target for an extension of the asymptotic-preserving procedure
given in [1]. Thanks to it we are able to establish the preservation of the asymptotic and the set of admissible
states under a classical hyperbolic-type CFL condition.
∗ Correspondence
27
Theorem 1. Let us consider the Rusanov-like following numerical flux for the i-th edge of the mesh:
Fi .ni = F̄i .ni −
bi θi
∇i U.ni ,
2
where:
• ∇i U.ni is Droniou and Le Potier’s approximation of the normal gradient of U on the i-th edge,
• F̄i .ni = ∑ λi, j
j
F(UK )+F(U j )
.n j
2
is a consistent approximation of F(U).ni such that λi, j ≥ 0,
• bi is greater than all waves speeds on the i-th edge, and θi = θi (U) ≥ 0 depends on the mesh.
Using this flux along with the technique introduced in [1] leads to a numerical scheme which:
• degenerates into Droniou and Le Potier scheme for (2) in the asymptotic regime,
• preserves the set of admissible states under the following CFL condition:
max η j
j,K
∆t 1
≤ ,
δK
2
where δK is a characteristic length of the cell K and η j = η j (bi , θi , λi, j ).
Finally, high-order extensions may be considered using classical techniques, for instance MOOD or MUSCL,
due to the form of the scheme.
REFERENCES
[1] C. Berthon, P. Le Floch, and R. Turpault. Late-time/stiff relaxation asymptotic-preserving approximations
of hyperbolic equations. Math. Comp. 82, pp 831-860, 2013.
[2] C. Buet, B. Després and E. Franck. Design of asymptotic preserving finite volume schemes for the hyperbolic heat equation on unstructured meshes. Numer. Math., 122(2) pp 227-278, 2012.
[3] J. Droniou and C. Le Potier. Construction and convergence study of schemes preserving the elliptic local
maximum principle. SIAM J. Numer. Anal., 49(2) pp 459-490, 2011.
[4] C. Sarazin-Desbois. Méthodes numériques pour des systèmes hyperboliques avec terme source provenant
de physiques complexes autour du rayonnement. Thèse de doctorat de l’Université de Nantes, 2013.
28
VERY HIGH ORDER FINITE VOLUME SCHEME FOR THE 2D AND 3D
LINEAR CONVECTION DIFFUSION EQUATION
S. Clain a,b∗ , G.J. Machado a , R.M.S. Pereira a , A. Boularas c
a
Departamento de Matemática e Aplicações e Centro de Matemática, Campus de Gualtar - 4710-057 Braga, Portugal.
Institut de Mathématiques de Toulouse, Université de Toulouse 31062 Toulouse, France. c Laplace Centre, Paul
Sabatier University, 31062 Toulouse, France
b
ABSTRACT
A sixth-order finite volume method is proposed to solve the bidimensional linear steady-state convectiondiffusion equation. A new class of polynomial reconstructions is proposed to provide accurate fluxes for
the convective and the diffusive operators. The method is also designed to compute accurate approximations even with discontinuous diffusion coefficient or velocity and remains robust for large Péclet
numbers. Discontinuous solutions deriving from the linear heat transfer Newton law are also considered
where a decomposition domain technique is applied to maintain an effective sixth-order approximation.
We also propose a new technique to accuratly take a curved boundary into account maintaining the sixthorder of convergence and the method is extended to the three-dimensional case. For more details see
[1, 2]
FRAMEWORK
Let Ω be a bounded polygonal domain of R2 or R3 and ∂ Ω its boundary. We consider situations where
the diffusion and the convection functions may comprise discontinuities, so domain Ω is partitioned in two
subdomains Ω1 and Ω2 sharing a common interface Γ:
∇.(V1 φ1 − k1 ∇φ1 ) = f1 ,
∇.(V2 φ2 − k2 ∇φ2 ) = f2 ,
in Ω1 ,
(1a)
in Ω2 ,
(1b)
k1 ∇φ1 .nΓ = k2 ∇φ2 .nΓ ,
on Γ,
(1c)
φ1 = φ2
on Γ,
(1d)
φ = φD ,
on ΓD ,
(1e)
− k∇φ .n = gP ,
on ΓP ,
(1f)
on ΓT ,
(1g)
V.nφ − k∇φ .n = gT ,
with adequate boundary conditions.
Applying the generic finite volume procedure and using the Gauss quadrature formula with R points on the
edges the sixth-order approximation writes
Gi =
|ei j | R
∑ ζr Fi j,r − fi ,
|ci | r=1
j∈ν(i)
∑
h
where Fi j,r stands for the numerical flux approximation of the physical flux function V (qi j,r ).n(qi j,r )φ (qi j,r )−
i
k(qi j,r )∇φ (qi j,r ).n(qi j,r ) evaluated at the Gauss point qi j,r .
∗ Correspondence
29
MAIN ISSUES
The sixth-order finite volume method we propose is based on the following ingredients.
• We introduce a new class of polynomial reconstructions associated to cells for the convective part or
edges/faces for the diffusive contribution. Conservative and non conservative reconstructions will be
proposed based on a fuctional minimization where a weighted strategy provides the positivity preserving
property for pure diffusive problems.
• We include a new technique to prescribe boundary conditions for curved domains up to the sixth-order
deeply based on the reconstruction procedure we have adopted.
• Another issue concerns the treatement of discontinuous coefficients and even discontinuous solutions
still preserving a sixth-order accuracy.
• The method is matrix free and the linear system is solved using a preconditionning GMRES method.
We present a new and simple preconditionning matrix based on the low-order "Patankar" discretization
and a pseudo inverse procedure which dramatically reduces the number of iterations and save o lot of
computational resources.
NUMERICAL RESULTS
Numerical tests to assess the scheme capacity to handle classical situations have been proposed such as low
and large Péclet number problems, pure convective problem, discontinuous situations both for the coefficients
and the solutions. We also propose a concrete simulation deriving from polymer flow simulation.
FIGURE 1: Numerical simulation of a cooler with discontinuous coefficients
REFERENCES
[1] S. Clain, G. Machado, J. M. Nóbrega, R. Pereira, A sixth-order finite volume method for multidomain
convection-diffusion problem with discontinuous coefficients. Computer Methods in Applied Mechanics
and Engineering, 267, (2013), 43–64.
[2] A. Boularas, S. Clain, F. Baudoin, A sixth-order finite volume method for diffusion problem with curved
boundaries. Under preparation.
30
COMPACTNESS OF SEQUENCES OF APPROXIMATE SOLUTIONS OF
PARABOLIC EQUATIONS
T. Gallouët a∗ ,
a
I2M, Université d’Aix-Marseille, 13453 Marseille cedex 13, France.
ABSTRACT
We present some compactness results useful for proving the existence of solutions for some parabolic
equations by passing to the limit on approximate solutions. A main example is the case where these
approximate solutions are obtained with Finite Volume schemes.
COMPACTNESS RESULTS
The main compactness results for proving existence of a solution for some nonlinear parabolic PDE are
due to J. L. Lions (in an hilbertian framework), J. P. Aubin (in the case of L p -spaces, 1 < p < +∞) and J.
Simon (in the case of L1 -spaces). These results are proven with the compactness theorem of Kolmogorov (for
L p -functions taking values in an Hilbert or Banach space).
When the parabolic PDE is discretized by a numerical scheme (with a space-time discretization), in order to
prove the convergence of the approximate solution to the exact solution (as the discretization parameters tend to
0), a possibility is to use a discrete version of these compactness results, using a family of approximate spaces.
For some problems (as, for instance, the diffusion equation with the p-laplacian) it is sufficient to use a discrete
version of the so-called Aubin-Simon theorem. For some other cases, such as the Stefan problem, we have to
use a discrete version of the Kolmogorov theorem itself.
We first give a version of the Kolmogorov theorem using a sequence of subspaces of a Banach space B. Let
B be a Banach space and (Xn )n∈N be a sequence of Banach spaces included in B. We say that the sequence
(Xn )n∈N is compactly embedded in B if all sequence (un )n∈N such that un ∈ Xn (for all n ∈ N) and (kun kXn )n∈N
bounded is relatively compact in B.
Time compactness with a sequence of subspaces. Let 1 ≤ p < +∞, T > 0. Let B be a Banach space and
(Xn )n∈N be a sequence of Banach spaces compactly embedded in B. Let ( fn )n∈N be a sequence of L p ((0, T ), B)
satifying the following conditions
• The sequence ( fn )n∈N is bounded in L p ((0, T ), B).
• The sequence (k fn kL1 ((0,T ),Xn ) )n∈N is bounded.
• There exists Ra nondecreasing function η from (0, T ) to R+ such that limh→0+ η(h) = 0 and, for all h ∈
(0, T ) and n ∈ N, 0T −h k fn (t + h) − fn (t)kBp dt ≤ η(h).
Then, the sequence ( fn )n∈N is relatively compact in L p ((0, T ), B).
We now give a version of the Aubin-Simon theorem using a sequence of subspaces of B. Let B be a Banach
space, (Xn )n∈N be a sequence of Banach spaces included in B and (Yn )n∈N be a sequence of Banach spaces. We
say that the sequence (Xn ,Yn )n∈N is compact-continuous in B if the following conditions are satified
• The sequence (Xn )n∈N is compactly embedded in B.
• Xn ⊂ Yn (for all n ∈ N) and if the sequence (un )n∈N is such that un ∈ Xn (for all n ∈ N), (kun kXn )n∈N
bounded and kun kYn → 0 (as n → +∞), then any subsequence converging in B converge (in B) to 0.
∗ Correspondence
31
Aubin-Simon Theorem with a sequence of subspaces and a discrete derivative. Let 1 ≤ p < +∞. Let
B be a Banach space, (Xn )n∈N be a sequence of Banach spaces included in B and (Yn )n∈N be a sequence of
Banach spaces. We assume that the sequence (Xn ,Yn )n∈N is compact-continuous in B.
Let T > 0 and (un )n∈N be a sequence of L p ((0, T ), B) satifying the following conditions
• For all n ∈ N, there exists N ∈ N? and k1 , . . . , kN in R?+ such that ∑Ni=1 ki = T and un (t) = vi for t ∈ (ti−1 ,ti ),
i ∈ {1, . . . , N}, t0 = 0, ti = ti−1 + ki , vi ∈ Xn . (Of course, the values N, ki and vi are depending on n.)
i−1
Furthermore, we define a.e. the function ðt un by setting ðt un (t) = vi −v
for t ∈ (ti−1 ,ti ).
ki
p
• The sequence (un )n∈N is bounded in L ((0, T ), B).
• The sequence (kun kL1 ((0,T ),Xn ) )n∈N is bounded.
• The sequence (kðt un kL p ((0,T ),Yn ) )n∈N is bounded.
Then there exists u ∈ L p ((0, T ), B) such that, up to a subsequence, un → u in L p ((0, T ), B).
With the hypotheses of the previous theorems, another interesting question is to prove an additional regularity for u, namely that u ∈ L p ((0, T ), X) where X is some space closely related to the Xn (and included in B).
We now precise the meaning of the sentence “X closely related to the Xn ” and we give a regularity result.
Let B be a Banach space, (Xn )n∈N be a sequence of Banach spaces included in B and X be a Banach space
included in B. We say that the sequence (Xn )n∈N is B-limit-included in X if there exist C ∈ R such that if u is the
limit in B of a subsequence of a sequence (un )n∈N verifying un ∈ Xn and kun kXn ≤ 1, then u ∈ X and kukX ≤ C.
Regularity of the limit. Let 1 ≤ p < +∞ and T > 0. Let B be a Banach space, (Xn )n∈N be a sequence of
Banach spaces included in B and B-limit-included in X (where X is a Banach space included in B).
Let T > 0 and, for n ∈ N, let un ∈ L p ((0, T ), Xn ). We assume that the sequence (kun kL p ((0,T ),Xn ) )n∈N is
bounded and that un → u in L p ((0, T ), B) as n → +∞. Then u ∈ L p ((0, T ), X).
APPLICATION TO NUMERICAL SCHEMES
We consider here the case of a Finite Volume method with one unknown per cell and per time. With the
notations of the previous section, the B space is the space L p (Ω) with (for instance) p = 1 or 2. We have a
sequence of spatial meshes, (Tn )n∈N . For a given n, the approximate space, Hn , is the set of functions constant
on each cell. We set Xn = Yn = Hn . If u ∈ Hn , the Xn -norm is (for some q) a discrete W01,q -norm which reads,
using some quite natural notations, for 1 ≤ q < +∞,
kukq1,q,Tn =
∑
σ ∈Eint ,σ =K|L
mσ dσ |
uK − uL q
uK
| +
∑ mσ dσ | dσ |q
dσ
σ ∈Eext ,σ ∈EK
and, for q = ∞, kukq1,∞,Tn = max{Mi , Me , M} with
Mi = max{
|uK − uL |
|uK |
, σ ∈ Eint , σ = K|L}, Me = max{
, σ ∈ Eext , σ ∈ EK }, M = max{|uK |, K ∈ Tn }.
dσ
dσ
The Yn -norm is a dual norm. For r ∈ [1,R ∞], k·k−1,r,Tn is the dual norm of the norm k·k1,q,Tn with q = r/(r − 1).
That is, for u ∈ Hn , kuk−1,r,Tn = max{ Ω uv dx, v ∈ Hn , kvk1,q,Tn ≤ 1}.
REFERENCES
[1] E. Chénier, R. Eymard, T. Gallouët and R. Herbin. An extension of the MAC scheme to locally refined
meshes : convergence analysis for the full tensor time-dependent Navier-Stokes equations. Accepted for
publication in Calcolo, 2014.
[2] R. Eymard, P. Féron, T. Gallouët, C. Guichard and R. Herbin Gradient schemes for the Stefan problem.
Submitted 2013.
[3] T. Gallouët and J. C. Latché. Compactness of discrete approximate solutions to parabolic PDEs - Application to a turbulence model. Commun. Pure Appl. Anal. 11 (2012), no. 6, 2371-2391.
32
A HIGH ORDER FINITE VOLUME/FINITE ELEMENT PROJECTION
METHOD FOR LOW-MACH NUMBER FLOWS WITH TRANSPORT OF
SPECIES
A. Bermúdez a , S. Busto a , M. Cobas a , J.L. Ferrín a , L. Saavedra c , M.E. Vázquez-Cendón a∗
a
Departamento de Matemática Aplicada, Universidad de Santiago de Compostela 15782 Santiago de Compostela, Spain.
Departamento Fundamentos Matemáticos, Universidad Politécnica de Madrid E.T.S.I. Aeronáuticos. 28040 Madrid,
Spain.
b
ABSTRACT
The purpose of the work is to present a finite volume/finite element projection method for low-Mach
number flows in both viscous and inviscid cases.
Starting with a 3D tetrahedral finite element mesh of the computational domain, the momentum equation
is discretized by a finite volume method associated with a dual finite volume mesh where the nodes of
the volumes are the barycenters of the faces of the initial tetrahedra. These volumes have been already
used for the 2D shallow water equation (see [1]) and allow for an easy implementation of flux boundary
conditions. The transport-diffusion stage is explicit. Upwinding of convective terms is done by classical
Riemann solvers as the Q-scheme of van Leer or the Rusanov scheme (see, for instance, [7]).
Concerning the projection stage, the pressure correction is computed by a piecewise linear finite element
method associated with the initial tetrahedral mesh (see [4] and [5]). Passing the information from one
stage to the other is carefully made in order to get a stable global scheme (see [2]).
High order methods studied in [3] are analyzed and implemented to solve the transport-diffusion stage
for the convective and the diffusive terms. The numerical results are compared with those obtained with
professional software.
In the present work it is also included the transport equations of species and the results obtained using a
resolution coupled and uncoupled with the model of flows at low Mach number are analyzed. Moreover,
this analysis will allow us to implement in a efficient way a k − ε model.
We present some academic problems in order to analyze the order of convergence as well as several
classical test problems from fluid mechanics (see [6]). Numerical results are shown aiming to evaluate
the order of convergence of the method.
ACKNOWLEDGMENT
This project was co-financed with FEDER and Xunta de Galicia funds under the grant reference GRC2013014.
∗ Correspondence
33
34
Your notes
35
Local committee
S TÉPHANE C LAIN, Universidade
do Minho, Braga, Portugal.
J ORGE F IGUEIREDO, Universidade
G ASPAR M ACHADO, Universidade
R UI P EREIRA, Universidade do
Minho, Braga, Portugal.
Organizing/Scientific Committee
C HRISTOPHE B ERTHON, Université de Nantes, France.
S TÉPHANE C LAIN, Universidade do Minho, Braga, Portugal.
F RÉDÉRIC C OQUEL, Université Pierre et Marie Curie, Paris, France.
S TEVEN D IOT, Los Alamos National Laboratory, USA.
M ICHAEL D UMBSER, Università degli studi di Trento, Italy.
E NRIQUE F ERNÁNDEZ -N IETO, Universidad de Sevilla, Spain.
T HIERRY G ALLOUËT, Université de Marseille, France.
R APHAËL L OUBÈRE, Université de Toulouse, France.
C ARLOS PARÉS, Universidad de Málaga, Spain.
E LENA V ÁZQUEZ C ENDÓN, Universidade de Santiago de Compostela, Spain.
C AROLINA R IBEIRO, Universidade
Organizing Institutions
Centro de Matematica, Universidade do Minho, Braga, Portugal.
Institut Jean Leray, Université de
Nantes, France.
Institut de Mathématique de
Toulouse, Université de Toulouse,
France.
Sponsors: The organizers acknowledge the financial support of the Mathematical Institute of Toulouse (IMT),
University Toulouse III, l’Agence National pour la Recherche (ANR project “ALE INC(ubator) 3D”), and by FEDER
Funds through Programa Operacional Factores de Competitividade — COMPETE and by Portuguese Funds through
FCT — Fundação para a Ciência e a Tecnologia, within the Projects PTDC/MAT/121185/2010 and FCT-ANR/MATNAN/0122/2012 and ANR-12-IS01-0004 GeoNum

abstracts book!

Transcription

Similar documents

simulform - KnowTech3D

PDF (Author`s version) - OATAO (Open Archive Toulouse Archive

Contents of this book

22929-FEB DIN A4 12-Seiter englisch.indd

Demand

matrice_projet_3_ZI LA MALTERIE_anglais_Mise en page 1

Advanced Computational Engineering

status of the global human body models consortium (ghbmc)

Functional renormalisation group in a finite volume

Numerical Methods

iso 9001:2008 certified cad/cam/cae/cfd

Topolog´ıa Algebraica de Espacios Topológicos Finitos y Aplicaciones