sixth-order nite volume approximations for smooth curved ......sixth-order nite volume...

Sixth-order finite volume approximations forsmooth curved boundary domains.

Mathematical Center, School of Sciences, Minho University – Portugal

Stephane Clain, Ricardo Costa, Abderhamanne Boularas

This research was financed by FEDER Funds through Programa Operacional Factores de

Competitividade,COMPETE and by Portuguese Funds through FCT, Fundacao para a Ciencia e a

Tecnologia, within the Project PTDC/MAT/121185/2010, the project FCT-ANR/MAT-NAN/0122/2012 and

the strategic programme PEst-OE/MAT/UI0013/2014.

Ofir, 2015 May, 18th -22th

Trento, 2015 March, 16th -20th

Position of the problem

+ Substitution of smooth boundary domains with the polygonaldomain associated to the mesh provides at most a second-orderscheme.

â Very high-order approximations (third-order or more) require aspecific treatment of the boundary conditions.

â Techniques developed for finite difference, finite elements,Discontinuous Galerkin (isoparametric elements).

6 Very few things in the finite volume context (larger thanthird-order).


F. V. techniques for the boundary treatment

¬ Extra rows of ghost cells which are added beyond the geometricboundary of the computational domain (thick condition).

Enforce the boundary conditions by constraining theleast-squares reconstruction:- C. F. Ollivier-Gooch and M. Van Altena, A high-order accurate unstructured

mesh finite-volume scheme for the advection-diffusion equation, Journal of

Computational Physics, (2002).


What we propose

â Similar to C. F. Ollivier-Gooch and M. Van Altena: boundarycondition prescription via the Polynomial Reconstruction (PR).

The main difference

â O-V use the Gauss points on the localcurve to build the PR.

â We use the Gauss points on the edge

to build the PR and introduce a free pa-

rameter to prescribe the BC condition on

the Gauss points of the curve ”in the in-

tegral sense”.

b

b

b

b

b

b

v1

v2

qr

pr

c

b

bv1 v2

v3

b

b

b

c

qr

pr

+ Main advantage: PL independent of the curve, use the freeparameter to recover the H-O: a problem independent blackboxprocedure (we just need BC).


Generic H-O F.V. for convection diffusion

ż

Bci

pV.nφ´ k∇φ.nqds´ż

ci

f dx “ 0. (divergence theorem)

ci eij

cjqij,r

nij

ci cell, eij “ ci X cj edge.

nij normal vector iÑ j.

qij,r Gauss points on edge.

νpiq adjacent cells subset to ci.

ÿ

jPνpiq

|eij |Rÿ

r“1

ζr

”

V pqij,rq.nijφpqij,rq ´ kpqij,rq∇φpqij,rq.nijı

´|ci|fi “ Oph2Ri q. (Gauss Points)


Fluxes and residual formulation

Based on the previous expression: the residual formulation is

Gi “ÿ

jPνpiq

|eij |

|ci|

Rÿ

r“1

ζrFij,r ´ fi,

where

Fij,r « V pqij,rq.nijφpqij,rq ´ kpqij,rq∇φpqij,rq.nij ,

approximation of the flux at the Gauss point qij,r.

+ Sixth-order approximation for Fij,r.


P.R. for Boundary Condition

Prescribe Dirichlet on ΓD “ BΩ,eiD: edge on the boundary ,d: the polynomial degree,SpeiD, dq: the associated stencil,φiD: the free parameter,

pφiDpx; d, φiDq “ φiD `ÿ

1ď|α|ďd

Rd,αiD

!

px´miDqα ´Mα

iD

)

,

α “ pα1, α2q, miD the centroid of edge eiD

+ Set MαiD “

1

|eiD|

ż

eiD

px´miDqα ds to provide the

conservation.


Coefficients for pφiD

RdiD vector gathering coefficients Rd,α

iD

Assume mean values φ` on cells c`, ` P SpeiD, dq are known,

pRdiD minimizes the functional

EiDpRdiD; d, φiDq “

ÿ

`PSpeiD,dq

ωiD,`

” 1

|c`|

ż

c`

pφiDpx; d, φiDq dx´φ`

ı2,

with ωiD,` positive weights.

+ Coefficients ωiD,` are very important to provide”good”properties (M-matrix).


Polynomial reconstruction machinery and fluxes

for vector Φ “ pφiqi“1,...,I given

+ Several polynomial reconstructions

Conservative for cell ci: pφipΦq,

Non conservative for edge eij : rφijpΦq,

Conservative for boundary edge eiD: pφiDpΦ;φiDq.

+ Flux evaluations

Inner edges eij : Fij,r “rV pqij,rq.nijs

`pφipqij,rq`rV pqij,rq.nijs

´pφjpqij,rq´kpqij,rq∇rφijpqij,rq.nij

Boundary edge eiD: FiD,r “ rV pqiD,rq.niDs`pφipqiD,rq`

rV pqiD,rq.niDs´pφiDpqiD,rq´kpqiD,rq∇pφiDpqiD,rq.niD.

+ We use the geometry of the mesh! Not of the curve.


Resolution

¬ The polynomial reconstruction operators are linear. The flux computations are linear.® The residual expression is linear: Φ Ñ GipΦq.

We get a linear operator

Φ P RI Ñ GpΦq “ pG1pΦq, ..., GIpΦqq P RI .

+ Problem: Find Φ such that GpΦq “ 0.

* Matrix-free problem: use GMRES method.


Curved boundary treatment

‚ qiD,r Gauss points on eiD,

‚ piD,r Gauss points on thecurve,

‚ viD,1 viD,2 vertices of theedge,

‚ uiD,1 uiD,2 vertices of arc.

+ Edges and curves may not have common vertices


Free parameter evaluation

Define the boundary integral default on the piece of curve.

BiDpφiDq “Rÿ

r“1

ζr

´

φDppiD,rq ´ pφiDppiD,r;φiDq¯

.

Notice that φiD Ñ BiDpφiDq is affine: unique solution BiDpφ‹iDq

1 Compute HpφiDq and HpφiD ` 1q

2 Compute φ‹iD with

φ‹iD “ ´BiDpφiD ` 1qφiD ´BiDpφiDqpφiD ` 1q

BiDpφiD ` 1q ´BiDpφiDq.

+ Just need BC + Gauss Points and very fast.


Numerical Tests


The annulus

´∆φ “ 0

Constant Dirichletcondition

Rin “ 110, Rout “ 1.

Solution φpx, yq “a lnpx2 ` y2q ` b.


The annulus: convergence tables without (top) and with (bottom) corrections

Cell P1 P3 P5

804 9.26E-02 — 3,80E-02 — 5.10E-02 —2178 3.09E-02 2.0 9.46E-03 2.8 9.60E-03 3.08226 7.17E-03 2.2 2.09E-03 2.3 1.98E-03 2.4

24502 2.92E-03 1.7 8.05E-04 1.4 7.79E-04 1.3

Cell P1 P3 P5

804 6.40E-02 — 1.98E-02 — 2.40E-02 —2178 2.20E-02 2.2 1.00E-03 5.1 8.84E-04 6.68226 5.19E-03 2.2 4.87E-05 4.6 4.99E-06 7.8

24502 1.20E-03 2.1 3.60E-06 3.7 9.09E-08 5.9


The 3D annulus

´∆φ “ 0

Constant Dirichletconditions

Rin “ 12, Rout “ 1.

Solution φpx, yq “a

a

x2 ` y2 ` z2` b.


The 3D annulus: convergence tables with (top) and without (bottom)

Cell P1 P3 P5

2924 5.19E-01 — 6.41E-02 — 2.24E-01 —7917 3.02E-01 1.6 2.52E-02 2.8 8.77E-02 2.8

17901 1.96E-01 1.5 1.27E-02 2.5 4.39E-02 2.531976 1.50E-01 1.3 7.47E-03 2.7 2.33E-02 3.2

Cell P1 P3 P5

2924 4.17E-01 — 3.94E-02 — 1.55E-01 —7917 2.06E-01 2,1 1.07E-02 3.9 7.33E-03 9.2

17901 1.21E-01 1,9 3.50E-03 4.1 1.17E-03 6.831976 8.11E-02 2,1 1.72E-03 3.7 4.11E-04 5.4


The annulus: non matching mesh with the boundary

h

%h

b

b

b b

b


Non matching mesh test: convergence tables

comparison P1 case with and without conrrections

Cell % P1 with P1 without

1557 42 9.42e-04 — 4.03e-03 —3257 40 5.13e-04 1.6 2.83e-03 0.967083 38 2.13e-04 2.3 1.91e-03 1.00

13771 36 1.18e-04 1.8 1.42e-03 0.9130569 33 5.23e-05 2.0 9.00e-04 1.14

P2, P3, P5 cases

Cell % P2 % P3 % P5

1557 21 1.43e-04 — 19 4.06e-05 — 18 1.25e-05 —3257 20 5.36e-05 2.7 17 6.43e-06 5.0 13 1.03e-06 6.87083 18 1.73e-05 2.9 15 1.44e-06 3.9 12 7.23e-08 6.8

13771 18 8.61e-06 2.1 15 4.39e-07 3.6 11 1.23e-08 5.330569 18 2.15e-06 3.5 13 7.61e-08 4.4 10 1.08e-09 6.0


Other Non matching mesh test

-100% of h radius: 1 Ñ 0.86

Cell h P2 P3 P5

1460 1.4e-1 6.26e-04 — 6.48e-05 — 5.71e-05 —3120 9.7e-2 3.31e-04 1.7 1.56e-05 3.8 3.35e-06 7.57082 6.5e-2 1.50e-04 1.9 2.87e-06 4.1 2.51e-07 6.3

13506 4.7E-2 8.15e-05 1.9 7.54e-07 4.1 3.83e-08 5.8

+100% of h radius: 1 Ñ 1.14

Cell h P2 P3 P5

1460 1.4e-1 6.91e-04 — 3.43e-05 — 9.98e-06 —3120 9.7e-2 3.66e-04 1.7 8.09e-06 3.8 9.55e-07 6.27082 6.5e-2 1.56e-04 2.1 1.48e-06 4.1 6.89e-08 6.4

13506 4.7E-2 8.44e-05 1.9 4.82e-07 3.5 2.41e-08 —


Find the Gauss points on the curve

e “ v1v2 boundary edge of length |v1v2|

Ŋv1v2 boundary arc of length |Ŋv1v2|

equal-distance property characterizes the Gauss points

|Ŋv1pr|

|v1qr|“|Ŋv2pr|

|v2qr|“|Ŋv1v2|

|v1v2|

âQuadrature rule for the numerical integration over the arc

ż

Ŋv1v2φDppqdp “ |Ŋv1v2|

R2ÿ

r“1

ξrφDpprq.


Conclusions and perspectives

A simple ”blackbox”procedure with one free parameter

Reconstruction matrix structure independent of the curve

Enable to handle boundaries which not fit with the mesh

Preserve the optimal order

Work for more complex problem (Stokes + ∇ ¨ U “ 0)


Questions

How treat the 3D case? do we have the equal-area propertyfor surface?

How to treat the Neuman condition (flux and condition arenot at the same place)?

What about the hyperbolic case with (partial) Dirichletcondition?

Uniform meshes with complex boundaries can be tackled?

Fracture, crack, operator with coefficient discontinuitiesacross a smooth line?

Tracking discontinuities or interfaces up to the sixth-order?


sixth-order nite volume approximations for smooth curved ......sixth-order nite volume...

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