etude de l’ordre de convergence de la méthode des volumes …...etude de l’ordre de convergence...

Etude de l’ordre de convergence de laméthode des volumes finis

Frédéric PASCAL

[email protected]

CMLA, ENS de Cachan

Joint work with

Daniel BOUCHE and Jean-Michel GHIDAGLIA

CMLA, ENS de Cachan

Groupe de travail "Méthodes numériques" LJLL, 08/10/07

Etude de l’ordre de convergence de la méthode des volumes finis – p. 1/27

Supraconvergence

Definition :Supraconvergence is observed when the global error of a Finite Differences scheme appliedto an ODE or PDE has a better behavior than that indicated by the local error.

Examples :

Order 2 scheme with order 1 local error

Order 1 (therefore convergent) scheme with a non-zero convergent local error.

Consequences :Lax Theorem is useless

Applications :This loss of accuracy real for the local error and apparent for the global error is observedwith non uniform grids


Lax TheoremFinite differences scheme : (h −→ 0 : parameter of discretization, Lh assumed linear)

Lh(uh) = F

Stability : (c cst ind. on h) Truncation error : (u smooth exact sol.)

‖L−1h ‖ ≤ c Lh(u) = ǫh + F

Global error :

‖eh‖ := ‖uh − u‖ = ‖L−1h (Lh(eh))‖ ≤ c‖Lh(eh)‖ = c‖ǫh‖

⇓Lax Theorem :

If the scheme is consistant (i.e. if ‖ǫh‖ h→0−→ 0) then uhh→0−→ u with a rate at least equal to the

local error rate.


Diffusion in dimension 1

−u′′ = f with Dirichlet B.C.xxx xx

∆∆

i−1 i i+1

i i+1

0 N

FD Scheme : − 2ui−1

∆i(∆i + ∆i+1)+

2ui

∆i∆i+1− 2ui+1

∆i+1(∆i + ∆i+1)= f(xi) et C.B.

Local Error : ǫih =∆i − ∆i+1

3u′′′(xi) + O(h2) = O(h) for non uniform grid

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x

u

Solution Exacte (1.−x).*(atan(10.*(x−0.5))+atan(10*0.5))

10−4

10−3

10−2

10−1

100

101

102

103

104

h

Max

h /

Min

hMax h / Min h

10−4

10−3

10−2

10−1

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

h

Err

Inf

5≤ max h / Min h ≤ 4191

Erreur Inf2.198 log(h) + 0.819

Error in O(h2)

Keller, 78 : applied a FD scheme to a 1st order equiv. system =⇒ O(h2)

Manteuffel et White, 86 : discretization of the 1st order system =⇒ 2nd order eqwith Lh = D0D1, then ǫh = D1γ + ǫ =⇒ O(h2)


Mathematical Analysis

Supraconvergence :ǫh = O(hp−1) and numerically ‖eh‖ = O(hp)

Wendroff and White, Comp. Math. Appl., 89 :

=⇒ correction of the error for the mathematical analysis

if ǫh = O(hp−1)

but ǫh = Lh(γ) + ǫ

with γ = O(hp) and ǫ = O(hp)

then Lh(eh + γ) = −ǫ=⇒ ‖eh + γ‖ = O(hp)

=⇒ ‖eh‖ = O(hp)


Equation of Transport in dimension 1 (1/2)

−u′ + f = 0 and u(0) given

xxx xx

∆∆

i−1 i i+1

i i+1

0 N

xi−1/2 xi+1/21

FD Scheme : u0 given and − ui − ui−112(∆i + ∆i+1)

+ f(xi) = 0

Local error : ǫih =−∆i+1 + ∆i

∆i+1 + ∆iu′(xi) + O(h)= O(1) for non uniform grid

10−4

10−3

10−2

10−1

100

101

102

103

h

Max

i ∆i /

Min

i ∆i

Maxi ∆

i / Min

i ∆

i

10−4

10−3

10−2

10−1

10−5

10−4

10−3

10−2

10−1

h

||u−

u h|| ∞

||u−uh||∞

0.989 log(h) − 0.543

Numerical O(h) error


Transport in dimension 1 (2/2)

Correction :

ǫih =γi − γi−1

12(∆i + ∆i+1)

+ ǫi

with γi = −1

2∆i+1u

′(xi+1/2) = O(h)

and ǫi =∆i+1(u

′(xi+1/2) − u′(xi)) + ∆i(u′(xi) − u′(xi−1/2))

∆i+1 + ∆i= O(h)


Linear Convection Problem

a constant vector

Stationnary problem : 0 < b0 ≤ b(x) ≤ b1 :

8

<

:

bu+ (a · ∇)u = f in Ω

u(x) = ψ(x) on ∂Ω−

Unsteady problem :

8

>

>

>

<

>

>

>

:

∂u

∂t+ (a · ∇)u = 0 in Ω

u(x, 0) = φ(x) in Ω

u(x, t) = ψ(x, t) on ∂Ω− × [0,∞[


Finite Volume Method

Explicit, Finite volume scheme : unj ≈ 1

|Kj |

Z

Kj

u(x, tn)dx ≡ Unj such that 1 ≤ j ≤ Nv

un+1j − un

j

∆tn+

1

|Kj |

0

B

@

X

k∈N+(j)

a ·Nj,kunj +

X

k∈N−

0(j)

a ·Nj,kunk +

X

k∈N−

b(j)

a ·Nj,kφ(gj,k)

1

C

A= 0

nj,k1nj,k4

2j,kn

gj,k

gj,k

2

gj,k4

gj,k3

1gj

nj,k3

k1

k3

a2

k4

j

k

k

Nj,k = |Kj ∩Kk|nj,k

k1, k3 ∈ N+(j) and k2, k4 ∈ N−0 (j)

Hyp :hnd

j

|Kj |≤ κ1 and ♯N (j) ≤ κ2 , ∀Kj


Local and Global error

FV scheme :

(1) un+1j −Ln

j

`

unkk

´

+ Lnbord = 0 where

Lnj

`

ξkk

´

= ξj − ∆tn

|Kj |

0

B

@

X

k∈N+(j)

a ·Nj,kξj +X

k∈N−

0(j)

a ·Nj,kξk

1

C

A

Local error :

(2) u(gj , tn+1) −Ln

j

`

u(gk, tn)k

´

+ Lnbord = ∆tn ǫ

nj

Global error : enj = un

j − u(gj , tn)

⇒ (3) en+1j − Ln

j

`

enkk

´

= −∆tn ǫnj


Stability

Theorem 1 :

Under CFL∆tn

minj τj≤ 1 with τj =

|Kj |P

k∈N+(j) a ·Nj,k

i) ‖Ln(ξ)‖p ≤ ‖ξ‖p where ||ξ||p =

0

@

NvX

j=1

|Kj ||ξj |p1

A

1/p

and ||ξ||∞ = max1≤j≤Nv

|ξi|

ii) ‖en‖p ≤ ||e0||p +

n−1X

i=0

∆ti||ǫi||p

Lax Theorem :Local error estimate transfers to global error estimate


Consistency

Local error : ǫnj = Gnj + In

j + B. C. discretization error

• Error on B.C. discretization assumed small

• Centered part :

Gnj =

u(gj , tn+1) − u(gj , tn)

∆tn+

1

|Kj |X

k∈N (j)

a ·Nj,ku(gj,k, tn)= O(h)

• Upwind part :

Inj =

1

|Kj |

0

@

X

k∈N+(j)

a ·Nj,k

“

u(gj , tn) − u(gj,k, tn)”

+X

k∈N−

0(j)

a ·Nj,k

“

u(gk, tn) − u(gj,k, tn)”

1

C

A= O(1)

FVM is not consistent in the FD sense


Correction

Can we write ǫnj =1

∆tn

“

γn+1 −Lnj (γn)

”

+ ǫnj with γn ≡ γnj j = O(h) and ǫnj = O(h) ?

Geometric corrector : γnj = −Γj · ∇u(gj , t

n) where

Γ ≡ Γjj depends only on the mesh and vector a and is solution of

X

k∈N+(j)

a ·Nj,kΓj +X

k∈N−

0(j)

a ·Nj,kΓk =X

k∈N+(j)

a ·Nj,k(gj,k − gj) +X

k∈N−

0(j)

a ·Nj,k(gj,k − gk)

⇐⇒ (I −B)Γ = ∆ with (Bψ)j =

X

k∈N−

0(j)

a ·Nj,kψk

X

k∈N+(j)

a ·Nj,k


Existence of the geometric corrector

Theorem :σ(B) ⊂ z ∈ C , |z| < 1

et (I −B)−1 =∞X

i=0

Bi

Dependence cone : let J a volume of control

C(J) = K /∃ J1, · · · , Jn , a ·NJ1,K < 0, a ·NJ2,J1< 0, · · · , a ·NJ,Jn

< 0

a J

K

J1

J2 J3


Existence of the geometric corrector

Theorem :σ(B) ⊂ z ∈ C , |z| < 1

et (I −B)−1 =∞X

i=0

Bi

Proof :

a) ‖Bx‖∞ ≤ ‖x‖∞ sinceX

k∈N (j)

a ·Nj,k = 0

b) ∀J ∈ T , there is at least K ∈ C(J) sharing a face with ∂Ω−

Input Boundary plays an important role

c) By contradictionLet λ / |λ| = 1 et x / Bx = λx.

For Kj / |xj | = maxk

|xk|, we get N−b (Kj) = ∅ and ∀k ∈ N−

0 (j), |xk| = |xj |.

By induction, ∀K ∈ C(Kj), N−b (K) = ∅ contradiction with b)


Main result

Theorem : let γnj = −Γj · ∇u(gj , t

n) and let assume u smooth enough

i) local quasi-uniformity of the mesh1

κ3|Kk| ≤ |Kj | ≤ κ3|Kk| , ∀h < h0, ∀Kj ∈ T h, ∀k ∈ N (j)

ii) I.C.. : ||(u0j − ϕ(gj))j ||p ≤ κ4h ,

iii) B.C. : |unk − ψ(gj,k, tn)| ≤ κ5h2

then under CFL ,∀p ∈ [1,+∞], (α ∈]0, 1])

∃cp > 0 / ‖Γ‖p ≤ cphα =⇒ ‖u(·, tn) − (unj )j‖p ≤ cphα ∀tn ≤ T

Extension :

Implicit scheme

Mesh where interfaces are not hyperplane

The center of gravity gj can be replaced by any point at a distance less than h fromgj .


Study of the Geometric corrector (1)

Dimension 1 :

gj,j−1 gjxx jxj−1/2

gj,j+1j+1/2

e_jej∆ jx

a > 0

un+1j − un

j

∆tn+ a

unj − un

j−1

∆xj= 0

8

<

:

aΓ1 = a (x 32

− x1) , j = 1

a (Γj − Γj−1) = a (xj+ 12

− xj) − a (xj− 12

− xj−1) , j ≥ 2=⇒ Γj =

∆xj

2

Remark : γnj = −∆j

2

∂u(xj , tn)

∂x=⇒ en

j = unj − u(xj+1/2, tn) + O(h2)



Theorem :

−Dimension 2

−T0 unstructured coarse mesh of triangles or quadrangles

−Th obtained by uniformly refinement

then ‖Γ‖p ≤ ch

Proof :a) Start with one triangle : ♯T0 = 1

−−

−−

12

−+

−+

+−

−

+

−

2n

1

− +

m

T2,2−

2,2T+

T+1,2

+

2,1T+

+1

+1

a

+−+

++

+

+

+

−

−−

−

−+

12

12

++

m

n

T

T2,2

2,2

+

−

+1

+1

a

=⇒ |Γǫm,n| ≤

C(a, T )

ℓ+ 1



Theorem :

−Dimension 2

−T0 unstructured coarse mesh of triangles or quadrangles

−Th obtained by uniformly refinement

then ‖Γ‖p ≤ ch

Proof :b) By induction on the number of volumes

Lemma : convex volumes in 2d

∃ a broken line of interfaces that are “en-lighted” on the same side which devidethe domain in 2

A

A

A

A

A

S

S

S

S

S

F

44

4

3

3

32

2

2

1

I1

aP2

P3

n

nn

n



Numerical test with independent meshes :

a=0

a=pi/4

10−2

10−1

100

10−3

10−2

10−1

h

||Γ|| 1

θ = π/4

1.052 log(h) − 0.550

10−2

10−1

100

10−3

10−2

10−1

h

||Γ|| ∞

θ = π/4

0.905 log(h) − 0.469

10−2

10−1

100

10−3

10−2

10−1

100

h

Err

Err L ∞ 0.874 log(h) + 0.054 Err L1 0.988 log(h) − 0.593

10−2

10−1

100

10−3

10−2

10−1

h

||Γ|| 1

θ = 0

1.033 log(h) − 0.502

10−2

10−1

100

10−3

10−2

10−1

h

||Γ|| ∞

θ = 0

0.435 log(h) − 0.713

10−2

10−1

100

10−3

10−2

10−1

100

h

Err

Err L ∞ 0.441 log(h) − 0.427 Err L1 0.992 log(h) − 0.850

‖.‖1 ‖.‖∞

O(h)

‖.‖1 ‖.‖∞

O(h1/2)


Counter Example Peterson (Sinum 91)

T

TT

T

+

+

−

−

+

T−

T −T+

T+

T+

0,1

1,0 3,02,1

0,1 2,1

1,2

1,2

0,3

0,3 = 2 = 4

T

a

= 8

j

i

Numbering of triangles and definition of gj :

+ ++

m,n m,n

(T ,G )m,n m,n

(T ,G )

0,n

0,n

(T ,G )0,n

(T ,G )0,n− − −−

+

+

m+1,n−1

+ +

m−1,n−1 m−1,n−1(T ,G ) (T ,G )m+1,n−1

+



T

TT

T

+

+

−

−

+

T−

T −T+

T+

T+

0,1

1,0 3,02,1

0,1 2,1

1,2

1,2

0,3

0,3 = 2 = 4

T

a

= 8

j

i

Geometric Corrector :

Γ+m,n = 1

2(Γ+

m−1,n−1 + Γ+m+1,n−1) , m ≥ 1 , n ≥ 1

Γ+0,n = Γ

+1,n−1 + h

2~i , n = 1, 3, . . . , 2ℓ− 1

Γ+m,0 = 0 , m = 1, 3, . . . , 2ℓ− 1 .



T

TT

T

+

+

−

−

+

T−

T −T+

T+

T+

0,1

1,0 3,02,1

0,1 2,1

1,2

1,2

0,3

0,3 = 2 = 4

T

a

= 8

j

i

Explicit estimation equal to the analytical error estimation :

Γ+2p,2q+1 =

8

>

>

<

>

>

:

0 , 0 ≤ q ≤ ℓ2− 1 , q + 1 ≤ p ≤ ℓ− q − 1

qX

k=p

1

22k+1

“ 2k

k − p

”

h~i , 0 ≤ q ≤ ℓ− 1 , 0 ≤ p ≤ min(q, ℓ− q − 1)

||Γ(ℓ)||∞ =1√πh1/2 + O(h) et ||Γ(ℓ)||1 = O(h)


Counter Example Peterson:oblique incidence

T

TT

T

+

+

−

−

+

T−

T −T+

T+

T+

0,1

1,0 3,02,1

0,1 2,11,2

1,2

0,3

0,3 = 2 = 4

T

= 8

j

i

a

θ

Geometric corrector :

Γ+m,n = pΓ+

m−1,n−1 + qΓ+m+1,n−1 , m 6= 0 , n ≥ 1

Γ+0,n = q

pΓ

+1,n−1 + h~β , n = 1, 3, . . . , 2ℓ− 1

Γ+m,0 = 0 , m = 1, 3, . . . , 2ℓ− 1 ,

p =cosα

cosα+ sinα, p+ q = 1 , q < 1/2 < p , α =

π

4− θ , θ ∈]0, π/4] , ~β constant


Counter Example Peterson:oblique incidence

T

TT

T

+

+

−

−

+

T−

T −T+

T+

T+

0,1

1,0 3,02,1

0,1 2,11,2

1,2

0,3

0,3 = 2 = 4

T

= 8

j

i

a

θ

Results :

||Γ(ℓ)||∞ = O(h) et ||Γ(ℓ)||1 = O(h)

A matrix formulation (square of a matrix of transition of a Markov process)

A probabilist approach : a sum of random walk


Second order scheme (1d)

1st order :duj

dt+ a

uj − uj−1

∆xj= 0

Γj − Γj−1 =∆xj

2− ∆xj−1

2

2nd order :duj

dt+ a

fj+ 12

− fj− 12

∆xj= 0 with fj+ 1

2

= a

0

@uj +∆xj

2

uj+1 − uj−1

∆xj+ 12

+ ∆xj− 12

1

A

γj = −Γj∂2u(xj , t)

∂x2

8

>

>

>

>

<

>

>

>

>

:

Γ′j − Γ′

j−1 = ρj − ρj−1 with ρj =∆xj(∆xj−1 + ∆xj − ∆xj+1)

8

Γ′j = Γj + αj(Γj+1 − Γj−1) with αj =

∆xj

∆xj−1 + 2∆xj + ∆xj+1

Γ′j = O(h2) and ‖Γ‖p ≤ c‖Γ′‖p


Non constant vectora (1d)

A Riemann finite volume approach :

un+1j − un

j

∆tn+

1

∆xj(Φj+ 1

2

(un) − Φj− 12

(un)) = 0

with Φj+ 12

(un) =aj+ 1

2

(unj + un

j+1)

2− σj+ 1

2

aj+ 12

(unj+1 − un

j )

2

=σj+ 1

2

+ 1

2aj+ 1

2

unj −

σj+ 12

− 1

2aj+ 1

2

unj+1

= a+

j+ 12

unj − a−

j+ 12

unj+1

un+1j = un

j − ∆tn

∆xj

“

a+j+1/2

unj − a−

j+1/2un

j+1 − a+j−1/2

unj−1 + a+

j−1/2un

j

”

with aj+1/2 ≡ a(xj+1/2) = a+j+1/2

− a−j+1/2

and σj+ 12

= sign(aj+ 12

)


Analysis

Stability : Under∆tn

∆xj(a+

j+ 12

+ a−j− 1

2

) ≤ 1 then ‖Ln(ξ)‖p ≤ (1 + ∆tn‖ax‖∞) ‖ξ‖p

Consistency : ǫnj = Gnj + In

j

Gnj =

u(xj , tn+1) − u(xj , tn)

∆tn+f(xj+ 1

2

, tn) − f(xj− 12

, tn)

∆xj= O(∆t) + O(h)

Inj =

Φj+ 12

(Un) − Φj− 12

(Un) + f(xj− 12

, tn) − f(xj+ 12

, tn)

∆xj.

Corrector : γnj = u(xj , tn) − u(xj + σj

∆xj

2, tn) = −σj

∆xj

2

∂u(xj , tn)

∂x+ O(h2)

σj = sign(a(xj))

−1 −0.5 0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x

cvtM

, u

a(x) = x with Riemann Scheme

10−3

10−2

10−1

10−4

10−3

10−2

10−1

h

Err

Inf

a(x) = x with Riemann Scheme

Erreur Inf0.967 log(h) − 0.459

For γnj = −Γj

∂u(xj ,tn)

∂x,

the equation satisfied by Γj

has the same form as constanta but is no longer 0 but O(h2)

near a change of sign of a


Non constant vectora(x) (1d)

A flux scheme :

un+1j − un

j

∆tn+

1

∆xj

“

Ψj+ 12

(un) − Ψj− 12

(un)”

= 0

with Ψj+ 12

(un) =aju

nj + aj+1u

nj+1

2− σj+ 1

2

aj+1unj+1 − aju

nj

2

=σj+ 1

2

+ 1

2aju

nj −

σj+ 12

− 1

2aj+1u

nj+1

un+1j = un

j − ∆tn

∆xj

„

1 − σj+1/2

2(aj+1u

nj+1 − aju

nj ) −

1 + σj−1/2

2(aj−1u

nj−1 − aju

nj )

«ff

×aj

with σj+1/2 = sign(a(xj+1/2))

In term of fluxes : ϕnj = aju

nj satisfy

ϕn+1j − ϕn

j

∆tn+

aj

∆xj

1 − σj+ 12

2(ϕn

j+1 − ϕnj ) +

1 + σj− 12

2(ϕn

j − ϕnj−1)

!

= 0


Analysis

Stability on the flux : under∆tn

∆xjaj

σj+ 12

+ σj− 12

2≤ 1 then

‖LnF (ξ)‖p ≤ (1 + c∆tn‖ax‖∞) ‖ξ‖p

Consistency : ǫnj = Gnj + ajI

nj

Gnj =

f(xj , tn+1) − f(xj , tn)

∆tn+ aj

f(xj+ 12

, tn) − f(xj− 12

, tn)

∆xj= O(∆t) + O(h)

Inj =

Ψj+ 12

(Fn) − Ψj− 12

(Fn) + f(xj− 12

, tn) − f(xj+ 12

, tn)

∆xj.

Correction on flux : γnj = f(xj , tn) − f(xj + σj

∆xj

2, tn) = −σj

∆xj

2

∂f(xj , tn)

∂x+ O(h2)

−1 −0.5 0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

cvtM

, u

a(x) = x with Flux Scheme

10−3

10−2

10−1

10−4

10−3

10−2

h

Err

Inf F

lux

a(x) = x with Flux Scheme

Erreur Inf Flux0.964 log(h) − 0.824

Error estimate on the flux


The nonlinear case

Nonlinear problem :∂u(x, t)

∂t+∂f(u(x, t))

∂x= 0

Scheme :un+1

j − unj

∆tn+

1

∆xj(Φn

j+ 12

(un) − Φnj− 1

2

(un)) = 0

Φnj+ 1

2

(un) =f(un

j+1) + f(unj )

2− σn

j+ 12

f(unj+1) − f(un

j )

2

with σnj+ 1

2

= sign(snj+ 1

2

) where snj+ 1

2

=

8

>

<

>

:

f(unj+1) − f(un

j )

unj+1 − un

j

if unj+1 6= un

j

f ′(unj ) if un

j+1 = unj

Error analysis : enj = un

j − u(xj + δj∆xj

2, tn) with δj = sign(f ′(u(xj , ·))

Under ∆tn

∆xj

σn

j+ 12

−1

2snj+ 1

2

+σn

j− 12

+1

2snj− 1

2

!

≤ 1

||(enj )N

j=1||∞ ≤ C′∞h

||(enj )N

j=1||1 ≤ C′1h (for global quasi-uniform meshes)


Conclusion

Geometric corrector helps in finding optimal error estimates

Bounded domains and B.C. are taken into account

Lost of order of convergence for non smooth solution is not due to the mesh

Open question and Perspective :

Probabilist approach to study the corrector

Extension in 2d for non constant vector and 2nd order scheme

Study of the norm of the corrector is open in 2d


etude de l’ordre de convergence de la méthode des volumes …...etude de l’ordre de convergence...

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