Anti-dissipative schemes for advection and application to Hamilton-Jacobi-Bellman equations

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Anti-dissipative schemes for advection and application to Hamilton-Jacobi-Bellman equations

Olivier Bokanowski, Hasnaa Zidani

To cite this version:

Olivier Bokanowski, Hasnaa Zidani. Anti-dissipative schemes for advection and application to

Hamilton-Jacobi-Bellman equations. [Research Report] RR-5337, INRIA. 2004, pp.32. �inria-

00070664�

ISRN INRIA/RR--5337--FR+ENG

a p p o r t

d e r e c h e r c h e

THÈME 4

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Anti-dissipative schemes for advection and

application to Hamilton-Jacobi-Bellman equations

Olivier Bokanowski — Hasnaa Zidani

N° 5337

Octobre 2004

Unité de recherche INRIA Rocquencourt

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− V _j− ^n,R 1 2

,

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ν j := ∆t

∆x f (x j )

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ν j = 0

¬I&[foX~bn|utikjns_q [y

V _j ⁿ⁺¹ = V _j ⁿ

^{¬Fdgj q foXK[}

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V _j+ ^n,R/L 1 2

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ν j ν j+1 > 0

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= V _j+ ^n,R 1 2

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ν j ≥ 0 ∀j

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K

V _j ⁿ⁺¹ = V _j ⁿ − ν j

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2

.

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x _j

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V ∆x,∆t (t, x) = V _j ⁿ , x j−1/2 < x < x j+1/2 , t n ≤ t < t n+1 .

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R

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Od ¬Vxysikm˵/‡KµBAPâ¬/_äµ[ƒµ£¬

V _j ⁿ⁺¹ = V _j ⁿ − C _j− ¹

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2 (V _j+1 ⁿ − V _j ⁿ ), C _j− ¹

2 , D _j+ ¹

2 ∈ R ,

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j

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L ^∞

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.

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x ∈ R

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n

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b ⁺ _j

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0 < ν _j ≤ 1

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3

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j

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2

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K

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•

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m j+1/2 = min(V _j ⁿ , V _j+1 ⁿ )

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^{«¬/dƒj q}

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•

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:= V j+1

dƒj q

V _j+ ^n,L 1 2

:= V j .

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•

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ν j 6= 0

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V _j+ ^n,L 1 2

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2

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q

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q

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[»j[

q

_pjfhX[(u©dƒns[

ν _j+1 = 0

^µ

$à º–K•

@ @ S . Y *354[!(! T.4 ($! ,F:1R

ϕ ^{U B} (r, ν) := max(0, min( ^2r _ν , _1−ν ² ))

^:

•

ν j > 0

V _j+ ^n,L 1 2

= V j + ^1−ν ₂ ^j ϕ ^{U B} (r j , ν j )(V j+1 − V j )

^Y

r j = ^V _V ^{j −V j −1}

j+1 −V j

•

ν j < 0

V _j− ^{n,R 1}

2

:= V j + ^1−|ν ₂ ^{j |} ϕ ^{U B} (r _j ⁻ , |ν j |)(V j − V j+1 )

^{Y _}

r _j ⁻ := ^V _V ^{j+1 −V j}

j −V j −1 = _r ¹

j

•

ν j ≤ 0

^&

ν j+1 ≥ 0

^U

V _j+ ^n,R 1 2

:= V j+1

&

V _j+ ^n,L 1 2

:= V j

•

ν j ν j+1 > 0

^U=

V _j+ ^n,R 1 2

:= V _j+ ^n,L 1 2

ν j > 0

^!

V _j+ ^n,L 1 2

:= V _j+ ^n,R 1 2

ν j+1 < 0

$à º–K• @ @

ν j ≤ 0

^&

ν j+1 ≥ 0

^{' !$e ! T _ !& !e}

[

Y Y

& , R4#

4' ((\*!

Y

= f (). _

š [»/ysnsflnsfodgfo[#d0nh_pZCmKca[‹ys[tnhbKcpf©µ

• ÇÇ ’;>”>;Ç&? @ @ & [ ^

!& $!

|ν j | ≤ 1

∀j

(i)

^{=:1R .!}

(ii)

^{=:1R S}

L ^∞

^$$5(

• ÇËÇ @

(i)

^Jߥ

ν j > 0

^{dƒj q}

ν j+1 > 0

fhX[tj1['X/d©¤k[

V _j+ ^{L 1}

2

∈ [m _j+ ¹

2 , M _j+ ¹

2 ]

Œ‰rutiƒjnh_pnsfo[jur iƒ¥PfoX['´|² nhuX[ZC[g¬Fdƒj q dƒcpnhi

V _j+ ^{R 1}

2

= V _j+ ^{L 1}

2

Œ‰r q [»/j_£fo_aiƒj³iƒ¥“foX['´|²z  nhuXK[tZ8[ƒµ5W5X['u©dƒns[

ν j < 0

^{dƒj q}

ν j+1 < 0

_pnPns_aZ8_acedgytµ Jžj0foX[1u©dƒns[

ν j ≤ 0

^{dƒj q}

ν j+1 ≥ 0

¬ƒ&[5X/d©¤k[*fhX[*uikjnh_pnsfh[tjur

Œ‰r q [»/j_£fo_pikj § ƒŽ «wµJßfys[tZCdƒ_ajnfoi-nsfob q r³fhX[%utdƒnh[

ν j ≥ 0

^{dƒj q}

ν j+1 ≤ 0

^{µ š} [%X/d©¤ƒ[%[t_£foX[y

ν _j > 0

^{dƒj q}

V _j+ ^L 1 2

∈ [m _j+ ¹

2 , M _j+ ¹

2 ]

Œ‰r(uikjnsfhyhbufo_pikjˬƒiƒy

ν _j = 0

^{dƒj q} _aj(foXK_anu©dƒns[&jKiœuikj q _£fo_aiƒj _pnyh[¿~b_ays[ q ikj

V _j+ ^L 1 2

µXJžj fhX[-nhdƒZ8[C1dMrƒ¬[t_£foX[y

ν _j+1 < 0

^{dgj q}

V _j+ ^R 1 2

∈ [m _j+ ¹

2 , M _j+ ¹

2 ]

^Œ‰r

uikjnsfhyhbufo_pikjˬ/iky

ν j+1 = 0

^{dƒj q} _ajfhX_an5utdƒnh[#jiutikj q _pfh_aikj_pn5yh[t¿~b_pyh[ q ikj

V _j+ ^R 1 2

µ

(ii)

^Jߥ

ν j > 0

¬Ënh_pjut[%&[0X/d©¤k[

V _j+ ^L 1 2

∈ [b ⁺ _j , B _j ⁺ ]

^{dƒj q}

V _j− ^R 1 2

∈ [m _j− ¹

2 , M _j− ¹

2 ]

¬Ë&[0ikŒKfodƒ_aj

V _j ⁿ⁺¹ ∈ [m _j− ¹

2 , M _j− ¹

2 ]

^{µ W/iky}

ν j < 0

¬foX[#dƒyskbZ8[tj‰fonldƒys[(fhX[(nodƒZ8[ƒµ WP_pj/dƒcac£rk¬A_p¥

ν j = 0

^foXK[tj

V _j ⁿ⁺¹ = V _j ⁿ

µz¼œ[tjKut[(1[#ikŒKfodƒ_aj$fhX[

L ^∞

nsfodƒŒ_acp_pfžrC_ajdƒcpcËu©dgnh[tnµ

$à º–K•

@ @

C= *! ,^! C($\ =

L ^∞

^{5()] 5$f*}

5] !4[. ^(). ! ^=

4 V! ((

j

^{+ ! & _$! Y}

ν j ≤ 0

&

ν j+1 ≥ 0

^{f ()., _ !}

V _j+ ^R 1 2

&

V _j+ ^L 1 2

_ f* [* R! ^

L ^∞

^{$5()] _4 ()U}

! f] !& $! (!f ^4 $

[ %=[.*!

Y !4 ( $ # !

V _j+ ^R 1 2

&

V _j+ ^L 1 2

] !()4 Y

V _j

^&

V _j+1

^{F! b [ ,."}

L ^∞

^$$5(S

ÀœiM &[|fobysj$ikj$nªfob q _ajfhX[œfhiƒfdgcI¤gdƒys_edgfh_aikjiƒ¥

V ⁿ

^{µ š} [|nsX/dƒcacInod©r8foX/d!f

x ^∗

_pn*d'uyh_£fo_autdƒc mIik_aj‰fiƒ¥

f

^_p¥

f (x ^∗ ) = 0

^µ

ÄÇ ” à ’;<’ (( T$*(5!

x ^∗

^,

f

^{_V4 = =6}

f

^,M!

f! ! b !5 ,

x ^∗

^!

f

^{[V A}

x ^∗

^{:= _R h=}

> 0

^,

f |[x ^∗ −,x ^∗ [ >

0,

^&

f |]x ^∗ ,x ^∗ +] < 0

^!

f |[x ^∗ −,x ^∗ [ < 0,

^&

f |]x ^∗ ,x ^∗ +] > 0

4 T T! !()]

U 4 , *! ] 4[

• ÇÇ ’;>”>;Ç&?

@ @ 4

V ⁰

^M4

T V (V ⁰ ) < ∞

f

^{()[ !4 !4#&}

5M - 0&

!& $!

|ν j | ≤ 1

^{: = :1R} ',!4[& [*

! (U! T$! =R!(( Y ' C h !$

C ≥ 0

^{& [ & #,}

X& ,

V ⁰

^.4

∀n ≥ 0

T V (V ⁿ ) ≤ T V (V ⁰ )(1 + C∆t).

^{§ä EM«}

$à º–K•

@ @ : [$*( !4 &$! - " # [R+! b [ = 4[ S,R& [

j

^{4 C=}

ν j < 0

^&

ν j+1 > 0

^{!4 & [* f [ Y (( .!()] &([ Y}

! ( ! T$!F,^^ ] f *!

• ÇËÇ

@ U '! [ e *!

Y

ν j < 0

^&

ν j+1 > 0

^{f X#}

.4[() f!4#()] !

∆x

^{4 $()]X (( Jߥ}

ν _j ≥ 0

^{¬bKnh_ajK foX[}

L ^∞

 žnªfdƒŒK_aca_£fžr¶mysikmQ[ysfžr

&[0X/d©¤k[

V _j ⁿ⁺¹ = V _j ⁿ − C _j− ¹

2 (V j − V j−1 )

^_£foX

C _j− ¹

2 ∈ [0, 1]

^{¬Fdƒj q} &[0u©dƒjŠys_pfh[§ d «5_pfhX

D _j+ ¹

2 = 0

^µ Jžj¶foXK[8utdƒnh[X[yh[

ν j+1 ≤ 0

¬1[utdƒjºdgcanhiys_pfo[

V _j+1 ⁿ⁺¹

_aj¶foX[0¥¦ikysZ § d «œ_pfhX

C _j+ ¹

2 = 0

µ­¼œ[tjKut[#_ajdƒcacFu©dƒns[tn1[#X/d©¤k[(foX[(_pjutys[tZ8[tj‰fdƒcQ¥¦ikyhZ §d «&_pfhX

C _j+ ¹

2 + D _j+ ¹

2 ≤ 1

ns_aju[5ikj[*iĴ/fhX[*ui`[ Cu_a[j~f

C _j+ ¹

2

iky

D _j+ ¹

2

dƒc£*d©rAnP¤gdƒj_pnhX[ntµ š [*utiƒjutcpb q [1fhX/dgfUfhX[*nhuXK[tZ8[

_pnlW ‹µ

Y Y

'! [ "='*!

Y

f

^{[ T ! f # , !()]}

!f

h! !

> 0

^Y

f |[x ^∗ −,x ^∗ [ < 0,

^&

f |]x ^∗ ,x ^∗ +] > 0

D³ikyh[1myh[ut_ans[tc£rk¬ƒ&[

nsbmmIiknh[%XK[tyh[

ν j < 0

^{dƒj q}

ν j+1 > 0

§>_pfoX

ν j−1 ≤ 0

^{dƒj q}

ν j+2 ≥ 0

^¥¦iƒy

∆x

nhZCdƒcpcË[tjiƒbkXV«wµ

²1[u©dƒbKnh[ig¥“fhX[

L ^∞

nsfodƒŒ_pca_pfžrƒ¬ q [jiƒfh_aj

∆V _j+ ¹

2 := V j+1 − V j

¬K1[#u©dgjys_pfo[

V _j−1 ⁿ⁺¹ = V _j−1 ⁿ + D _j− ¹

2 ∆V _j− ¹

2

§âkƒd‰«

V _j ⁿ⁺¹ = V _j ⁿ + D _j+ ¹

2 ∆V _j+ ¹

2

§âkgŒV«

V _j+1 ⁿ⁺¹ = V _j+1 ⁿ − C _j+ ¹

2 ∆V _j+ ¹

2

§âkguM«

V _j+2 ⁿ⁺¹ = V _j+2 ⁿ − C _j+ ³

2 ∆V _j+ ³

2

§âk q «

_£foXutiA[ 8ut_p[tj‰fon

C, D

^_aj

[0, 1]

^{µ š} [(fhX[tjikŒKfodƒ_ajfhX[(caiAu©dƒcQŒQikbKj q

|∆V _j− ^{n+1 1}

2

| + |∆V _j+ ^{n+1 1}

2

| + |∆V _j+ ^{n+1 3}

2

| ≤ (1 − D _j− ¹

2 )|∆V _j− ^{n 1}

2 | + (1 − C _j+ ³

2 )|∆V _j+ ^{n 3}

2 | +

|1 − C _j+ ¹

2 − D _j+ ¹

2 | + C _j+ ¹

2 + D _j+ ¹

2

|∆V _j+ ^{n 1}

2 |.

^{§âƒ‡‰«}

JžjŠfoX[u©dƒns[

C _j+ ¹

2 + D _j+ ¹

2 ≤ 1

foXK[%nhuX[ZC[0_pn‹W Œ‰r ¼|dƒyªfo[tjF°±nuyh_pfh[tys_edKµ(^AbmmIiknh_pj

fhX/dgf‹&[0dƒys[%_aj fhX[%Œ/d q nh_pfhb/dgfh_aikj

T V (V ⁿ⁺¹ ) > T V (V ⁿ )

§<igfoX[ys_ans[foXK[tyh[_an‹jigfoX_pjCfhi mysiM¤k[M«w¬1[‹fhX~bn5X/d©¤k[

C _j+ ¹

2 + D _j+ ¹

2 > 1

^{dƒj q [_pfoXK[ty}

C _j+ ¹

2 > 1/2

^iky

D _j+ ¹

2 > 1/2

^{µ š [dƒcansi}

ikŒ`fdƒ_pj¥¦ysikZ § ƒ‡ «&fhX[¥¦ikcacpi—_pj%ŒIikbj q

T V (V ⁿ⁺¹ ) ≤ T V (V ⁿ ) + 2|∆V _j+ ^{n 1}

2 |.

^§âg\~«

W5X[tys[(yh[tZCdƒ_pjn*foiŒIikbj q

|∆V _j+ ¹

2 |

^{µz²&r q} [»/j_£fo_aiƒjig¥

V _j+1 ⁿ⁺¹

¬/bns_aj³§ kgu «w¬

V _j+ ^R 1 2

= V _j+1

^¬

dƒj

q

V ^L _j+ 3 2

= V _j+1 + ^1−ν ₂ ^j+1 ϕ _j+1 (V _j+2 −V _j+1 )

^X[yh[

ϕ _j+1 = ϕ ^{U B} (r _j+1 , ν _j+1 )

Œ‰r#‚l[Z-dƒybc ‡KµE µ

š [(ikŒKfdg_aj

C _j+ ¹

2 = ν j+1



 V _j+ ^n,L 3

2

− V _j+ ^n,R 1 2

V _j+1 − V _j



 = 1

2 ν j+1 (1 − ν j+1 ) ϕ j+1

r _j+1 .

^§âk†k«

[f‹bn|nsbmmIiknh[%fhX/dgf

C _j+ ¹

2 > 1/2

¬VfoX[iƒfoX[y|utdƒnh[§

D _j+ ¹

2 > 1/2

«5ŒQ[[t_ajK$nh_pZC_pcedƒyµ Jžj foX_pn utdƒnh[|&[|XdM¤ƒ[

r j+1 = ^∆V _∆V ^{j+ 1 2}

j+ 3 2

≤ 2ν j+1 ,

ŒI[tutdƒbnh[œiƒfoXK[tys_pnh[g¬Knh_pjut[

ϕ j+1 ≤ _1−ν ²

j+1

Œ~r § EB «¬~&[

&ikbcq X/d©¤ƒ[

C _j+ ¹

2 = 1

2 ν j+1 (1 − ν j+1 ) ϕ j+1

r _j+1 ≤ ν j+1

r _j+1 ≤ 1 2 .

Jžjm/dƒyªfo_aubcedgy

|∆V _j+ ¹

2 | ≤ 2ν j+1 |∆V _j+ ³

2 | ≤ 2ν j+1 max

k |∆V _k+ ^{n 1}

2 |.

Àli—›ns[fsfo_ajK

M ⁿ := | max k V _k ⁿ − min ` V <sub>`</sub> ⁿ |

¬&&[X/d©¤k[

|∆V _k+ ^{n 1}

2

| ≤ M ⁿ

^µ ´œns_aj¯foX[

L ^∞

nªfdƒŒ_pca_£fžrk¬&&[Šns[t[ fhX/dgf

M ⁿ ≤ M ⁰

^{µ É canhi¨1[ X/d©¤k[}

M ⁰ ≤ T V (V ⁰ )

^{µ ¼œ[tjKut[} &[ŠikŒKfodƒ_aj

|∆V _j+ ¹

2 | ≤ 2 max(|ν _j |, ν _j+1 )T V (V ⁰ )

µUWVbKysfoXK[tyhZ8ikys[|foX[yh[[wvK_pnsfhn

x ^∗ ∈ I _j = [x _j , x _j+1 ]

^nsbuX

fhX/dgf

f (x ^∗ ) = 0

^{µ|^`i}

|f (x)| ≤ L∆x

¥¦iky|dgcac

x ∈ I j

¬VX[tys[

L

_an|d-caiAu©dƒccp_amnsuX_pf*`'uikjnªfdƒj‰f©µ W5X~bnl1[(X/d©¤k[

max(ν _j+1 , |ν _j |) ≤ L∆t

^µ

É ffoX_pn0mQik_pj‰f%1[X/d©¤k[ikŒKfodƒ_ajK[ q

T V (V ⁿ⁺¹ ) ≤ T V (V ⁿ ) + C∆tT V (V ⁰ )

^_pfhX

C = 4L

^µ ¼œiM&[¤k[yt¬|fhX[¶u©dgnh[

T V (V ⁿ⁺¹ ) > T V (V ⁿ )

^_pj ¥<dƒufdgmmQ[tdƒyhniƒjcpr¹_aj d m/dgysfo_putbcadƒy uikjK»/kbKyodgfh_aikj¶iƒ¥­foXK[0nh[t¿~b[jut[

(V _j−1 ⁿ , V _j ⁿ , V _j+1 ⁿ , V _j+2 ⁿ )

^{dƒj q} &[8mIiknªfomIikj[%foi$fhX[8dƒmmI[tj q _£v fhX[(myhiAiƒ¥“iƒ¥“fhX[¥¦ikcacpiM_aj%ys[tnsbcpf

K

zÃ

º–

@

@ 4>#

f

^{[S !()]C!f & (}

j

"%& [ 4 =

ν _j < 0

^&

ν _j+1 > 0

T V (V ⁿ⁺¹ ) > T V (V ⁿ )

⁼

∀m ≥ n + 1

^Y

!4[&

T V (V ^m ) ≤ T V (V ⁿ⁺¹ )

¼œ[jut[_p¥fhX[nhuX[ZC[_an0jKiƒfW dgfnhiƒZC[$fh_aZ8[

t ⁿ

^{¬­_äµ[ƒµ}

T V (V ⁿ ) ≤ T V (V ⁿ⁻¹ ) ≤ T V (V ⁰ )

^{dƒj q}

T V (V ⁿ⁺¹ ) > T V (V ⁿ )

^foX[j &[¯nsfh_acac|X/d©¤k[¶¥¦iky

k ≥ n + 1

^¬

T V (V ^k ) ≤ T V (V ⁿ⁺¹ ) ≤ T V (V ⁿ ) +C∆tT V (V ⁰ ) ≤ T V (V ⁰ )(1 + C∆t)

¬~XK_auX8_anUfoXK[ q [tnh_pyh[ q ŒIikbj q µ

&(()] \+ *!

f

^[R

m ≥ 1

^{T _! f#0 &[}

ikŒ`fdƒ_pj³d0ns_aZ8_acedgy&ŒIikbj q _pfhX

C = 4mL

µ­|[fdƒ_pcandƒys[(ca[¥>f1foi0foX[(ys[©d q [ytµ

$à º–K• @ @ '*!

ν j < 0

^Y

C _j+ ¹

2 = ν j+1 V ^n,L

j+ 3 2

−V ^n,R

j+ 1 2

V j+1 −V j

4[ T

T!

ν j+1 > 0

V _j+ ^n,L 3 2

* 5] [U$! , = :1R

- f

V _j+ ^R 1 2

= V j+1

_

V ^L _j+ 1 2

= V j

=C& ( !$',

V _j+ ^R 1 2

.

[V j ; V j+1 ]

^..!

C _j+ ¹

2

_

D _j+ ¹

2

$à º–K•

@:@ 1R] # #&.$& # b 4[= ! *( ! !$! ( Y ",

Y .*+ Y .! [f ,^ 1R ^, Y * .!()4 $!F,

T4[f$!

v ∈ L ¹ _loc ((0, ∞) × R )

^{4 ! ((}

ϕ ∈ C ¹ ( R × R )

Z

R

v ₀ (x)ϕ(0, x)dx + Z

(0,∞)×R

vϕ _t + v f ⁰ ϕ + f ϕ _x ) dt dx = 0.

·/345

W5X[´œ²­  nhuXK[tZ8[%myh[nh[j~fh[ q _aj fhX[mKyh[¤A_pikbn‹ns[tufh_aikj¶ƒ[tj[yodƒcp_anh[n|ns_aZ8mcprfhX[0´œc£foyhd-²1[[

nsuX[tZ8[ƒµ&ÀœiM 1[œ*dƒj‰f5fhi%_aZ8myhiM¤ƒ[œfoXK[‹iky q [ty1ig¥fhX[‹´lcpfhyod! ž²1[t[œnhuX[ZC[gµ­Wi q i%nsi¬`&[bns[

d0¥¦ikyhZCdƒcp_anhZ„ns_aZ8_acedgy­fhifhX[(ikj[(iƒ¥U^A&[tŒ‰r

O

ƒŽ P¥¦iky*foXK[#jKikjutikjns[tyª¤ƒd!fo_p¤ƒ['[¿~b/dgfo_pikjÁ§ ‡ «wµ

A17ODOD1#1#%FA)( DOD#%7

WP_pyhnªf51[#utikjKnh_q [ty*foXK[#utdƒnh[#iƒ¥“mQiƒnh_pfh_p¤ƒ[‹¤k[tcpi`u_pfh_a[tn*dƒj q dƒnhnsbZC[fhX['MW Áuikj q _£fo_aiƒj

0 < ν _j ≤ 1.

š [(utikZ8[(Œ/dƒuc-fhifhX[(jikj` ßutikjKnh[tyª¤gdgfo_£¤k[¥¦ikyhZ § B «1_£foXd b`v q [»/j[ q Œ‰r

V _j+ ^{R 1}

2 = V _j+ ^{L 1}

2 = V _j+ ¹

2 := V j + 1

2 (1 − ν j )ϕ j (V j+1 − V j ), ϕ j ≥ 0.

^§äB‰«

š [cpi`ihc$¥¦iƒy|d8j[ ¥¦bjufo_pikj

ϕ j = ϕ ^{N B} (r j , ν j )

^_£foX

r j = ^V _V ^{j −V j−1}

j+1 −V j

¬QnsbuXŠfoX/d!f|foXK[nhuXK[tZ8[

ŒI[-W ‹¬

L ^∞

nªfdƒŒcp[ƒ¬Pdƒj q ig¥5iky q [ty`µ WViƒy(fhX_an¬“&[Cfodcƒ[-d¥¦bjufo_pikj

ϕ ^{N B}

foX/d!fnhdgfo_pns»/[n

0 ≤ ϕ ^{N B} ≤ ϕ ^{U B}

¬ns_aju[%foX_pn‹_aZ8mca_p[tnœW „dƒj q

L ^∞

nªfdƒŒK_aca_£fžrk¬FdƒnmyhiM¤ƒ[ q _aj

O† Päµ š [dgcanhi

_pZCmIiknh[

ϕ ^{N B} (1) = 1

^{_ajiky q} [tyUfhi(X/d©¤k[lnh[tuikj q iky q [tytµ š [5foXK[tj8uXi`iƒnh[5foX[5¥¦bjKufo_pikj q [»/j[ q

Œ‰r

K

ϕ ^{N B} (r, ν) = max(0, min(1, 2r

ν ), min(r, 2

1 − ν )).

^§äkˆk«

W5X[(¥¦bjufo_pikj

ϕ ^{N B}

_an5yh[myh[nh[j~fh[ q _pj WP_aµ E µ

š [-yh[tZCdƒybc¨fhX/dgf%¥¦iky

1

2 ≤ r ≤ 2

¬U&[X/d©¤k[

ϕ ^{N B} (r) = max(0, min(1, 2r), min(r, 2))

^¬

dƒn$‚liA[ƒ°n^`bKmQ[yª ¾²&[t[¶^AuX[tZ8[

O

ƒŽ PⵟW5X_pn-utdƒnh[Šuikyhys[tnsmQikj q n-foiÁnhZ8i`igfoX ns_pfhb/dgfo_pikjnµ W/iky

r ≤ ^ν ₂

^iky

r ≥ _1−ν ²

¬V&[XdM¤ƒ[

ϕ ^{N B} (r, ν) = ϕ ^{U B} (r, ν)

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|ν j | ≤ 1,

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V _j+ ^n,L 1 2

dƒj q

V _j+ ^n,R 1 2

Œ‰r

K

• V _j+ ^n,L 1 2

:= V j + 1

2 (1 − ν j )ϕ ^{N B} (r j , ν j )(V j+1 − V j ),

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ν j > 0

^{§âdkd‰«}

• V _j+ ^{n,R 1}

2

:= V j+1 + 1

2 (1 − |ν j+1 |)ϕ ^{N B} (r ⁻ _j+1 , |ν j+1 |)(V j − V j+1 ),

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ν j+1 < 0

^{§ädƒŒ/«}

• V _j+ ^n,L 1 2

= V _j+ ^n,R 1 2

:= 1

2 (V j + V j+1 )

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ν j+1 ≥ 0

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= V _j+ ^n,R 1 2

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ν _j ν _j+1 > 0

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r j := V j − V j−1

V j+1 − V j

dƒj q

r ⁻ _j+1 := V j+1 − V j+2

V j − V j+1

= 1/r j+1

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ν _j = 0

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ν _j+1 = 0

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š [(jiM k_p¤ƒ[nhikZ8[([tca[ZC[j‰fdƒyªr$mysikmQ[ysfh_a[tniƒ¥foXK_an5nhuXK[tZ8[

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ϕ _j+1 := ϕ ^{N B} (r _j+1 , ν _j+1 ) ≤ _2ν ^{r j+1}

j+1 ≤ ^r _ν ^j+1

j+1

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C _j+ ¹

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1

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L ^∞

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t = 1

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t = 1

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t = 5

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P = 50

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t = 5

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t = 1

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t = 5

dƒj q

P = 100

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N−BEE

1 2

2 1− ν

1 r

2r ν

2r

r

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ϕ(r)

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ϕ ^{N B}

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x

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t = 1