t rtrsts trt - - PowerPoint PPT Presentation

▼❡t❤♦❞ ♦❢ ❝❤❛r❛❝t❡r✐st✐❝s✿ t❤❡♦r❡t✐❝❛❧ ❛♥❞ ♥✉♠❡r✐❝❛❧ ❛♣♣❧✐❝❛t✐♦♥s

❨♦❤❛♥ P❡♥❡❧✶

✶❚❡❛♠ ❆◆●❊ ✭❈❊❘❊▼❆ ✕ ■♥r✐❛ ✕ ❯P▼❈ ✕ ❈◆❘❙✮

■♥ ❝♦❧❧❛❜♦r❛t✐♦♥ ✇✐t❤ ❙✳ ❉❡❧❧❛❝❤❡r✐❡ ✭❈❊❆ ❙❛❝❧❛② ✫ P♦❧②t❡❝❤♥✐q✉❡ ▼♦♥tré❛❧✮✱ ❏✳ ❏✉♥❣ ✭P❛✉✮

• ✳ ❋❛❝❝❛♥♦♥✐ ✭❚♦✉❧♦♥✮ ✫ ❇✳ ●r❡❝ ✭P❛r✐s ✺✮

❈❊▼❘❆❈❙ ✷✵✶✺

❈■❘▼ ✕ ✷✽ ❥✉✐❧❧❡t ✷✵✶✺

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s

❖✉t❧✐♥❡

✶

Pr❡s❡♥t❛t✐♦♥ ♦❢ Pr♦❥❡❝t ❉✐P▲♦▼❛

✷

■♥tr♦❞✉❝t✐♦♥ t♦ t❤❡ ♠❡t❤♦❞ ♦❢ ❝❤❛r❛❝t❡r✐st✐❝s

✸

❘❡s✉❧t✐♥❣ ❢❛♠✐❧② ♦❢ ♥✉♠❡r✐❝❛❧ s❝❤❡♠❡s

✹

❊①t❡♥s✐♦♥s ♦❢ t❤❡ ▼❖❈ s❝❤❡♠❡s

❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✷ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s Pr❡s❡♥t❛t✐♦♥ ♦❢ Pr♦❥❡❝t ❉✐P▲♦▼❛

❉✐P▲♦▼❛ ♦r✐❣✐♥s ❖♥❝❡ ✉♣♦♥ ❛ t✐♠❡ ✐♥ ✷✵✶✶✳✳✳

t❤❡ ✏❜❛s♠❛❝✑ ♣r♦❥❡❝t ✇❛s ♣r♦♣♦s❡❞ t♦ t❤❡ ✷✵✶✶ ❈❊▼❘❆❈❙ s❡ss✐♦♥✳ ■t ✇❛s t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ ❛♥ ❛♠❛③✐♥❣ r❡s❡❛r❝❤ ❛❝t✐✈✐t②✳ ■t ✇❛s ❛✐♠❡❞ ❛t st✉❞②✐♥❣ t❤❡ ❧♦✇ ▼❛❝❤ ♥✉♠❜❡r r❡❣✐♠❡✳ ❙♦ ✇❛s ❜♦r♥ t❤❡ ❈❉▼❆❚❍ ❣r♦✉♣✳

❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✸ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s Pr❡s❡♥t❛t✐♦♥ ♦❢ Pr♦❥❡❝t ❉✐P▲♦▼❛

▼♦❞❡❧s ❛♥❞ ♥✉♠❡r✐❝❛❧ ❛s♣❡❝ts

▲▼◆❈ ♠♦❞❡❧ ✭✈❡rs✐♦♥ ✷✵✶✶✮            ∇· ✉ = β(❤, ♣✵) ♣✵ Φ, ✭✶❛✮ ρ(❤, ♣✵) ×

• ∂t❤ + ✉ · ∇❤
• = Φ,

✭✶❜✮ ρ(❤, ♣✵) ×

• ∂t✉ + (✉ · ∇)✉
• − ∇· σ(✉) + ∇¯

♣ = ρ(❤, ♣✵)❣. ✭✶❝✮ ❊q✉❛t✐♦♥ ♦❢ st❛t❡✿ st✐✛❡♥❡❞ ❣❛s ❧❛✇ ❢♦r ❛ ♠♦♥♦♣❤❛s✐❝ ✢✉✐❞ ρ(❤, ♣✵) = γ γ − ✶ ♣✵ + π ❤ − q ❉✐♠❡♥s✐♦♥✿ ✶ ◆✉♠❡r✐❝❛❧ s❝❤❡♠❡✿ ▼❖❈ ✭▼❛t❧❛❜✮

❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✹ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s Pr❡s❡♥t❛t✐♦♥ ♦❢ Pr♦❥❡❝t ❉✐P▲♦▼❛

▼♦❞❡❧s ❛♥❞ ♥✉♠❡r✐❝❛❧ ❛s♣❡❝ts

▲▼◆❈ ♠♦❞❡❧ ✭✈❡rs✐♦♥ ✷✵✶✷✮            ∇· ✉ = β(❤, ♣✵) ♣✵ Φ, ✭✶❛✮ ρ(❤, ♣✵) ×

• ∂t❤ + ✉ · ∇❤
• = Φ,

✭✶❜✮ ρ(❤, ♣✵) ×

• ∂t✉ + (✉ · ∇)✉
• − ∇· σ(✉) + ∇¯

♣ = ρ(❤, ♣✵)❣. ✭✶❝✮ ❊q✉❛t✐♦♥ ♦❢ st❛t❡✿ st✐✛❡♥❡❞ ❣❛s ❧❛✇ ✇✐t❤ ♣❤❛s❡ ❝❤❛♥❣❡ ρ(❤, ♣✵) = γ(❤) γ(❤) − ✶ ♣✵ + π(❤) ❤ − q(❤) ❉✐♠❡♥s✐♦♥✿ ✶ ◆✉♠❡r✐❝❛❧ s❝❤❡♠❡✿ ■◆❚▼❖❈ ✭❋♦rtr❛♥✮

❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✹ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s Pr❡s❡♥t❛t✐♦♥ ♦❢ Pr♦❥❡❝t ❉✐P▲♦▼❛

▼♦❞❡❧s ❛♥❞ ♥✉♠❡r✐❝❛❧ ❛s♣❡❝ts

▲▼◆❈ ♠♦❞❡❧ ✭✈❡rs✐♦♥ ✷✵✶✸✮            ∇· ✉ = β(❤, ♣✵) ♣✵ Φ, ✭✶❛✮ ρ(❤, ♣✵) ×

• ∂t❤ + ✉ · ∇❤
• = Φ,

✭✶❜✮ ρ(❤, ♣✵) ×

• ∂t✉ + (✉ · ∇)✉
• − ∇· σ(✉) + ∇¯

♣ = ρ(❤, ♣✵)❣. ✭✶❝✮ ❊q✉❛t✐♦♥ ♦❢ st❛t❡✿ t❛❜✉❧❛t❡❞ ❧❛✇ ρ ∈ R✼[❤] ❉✐♠❡♥s✐♦♥✿ ✶ ◆✉♠❡r✐❝❛❧ s❝❤❡♠❡✿ ▼❖❈ ✭❋♦rtr❛♥✮

❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✹ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s Pr❡s❡♥t❛t✐♦♥ ♦❢ Pr♦❥❡❝t ❉✐P▲♦▼❛

▼♦❞❡❧s ❛♥❞ ♥✉♠❡r✐❝❛❧ ❛s♣❡❝ts

▲▼◆❈ ♠♦❞❡❧ ✭✈❡rs✐♦♥ ✷✵✶✹✮            ∇· ✉ = β(❤, ♣✵) ♣✵ Φ, ✭✶❛✮ ρ(❤, ♣✵) ×

• ∂t❤ + ✉ · ∇❤
• = Φ,

✭✶❜✮ ρ(❤, ♣✵) ×

• ∂t✉ + (✉ · ∇)✉
• − ∇· σ(✉) + ∇¯

♣ = ρ(❤, ♣✵)❣. ✭✶❝✮ ❊q✉❛t✐♦♥ ♦❢ st❛t❡✿ st✐✛❡♥❡❞ ❣❛s✴t❛❜✉❧❛t❡❞ ❧❛✇ ρ(❤, ♣✵) = γ(❤) γ(❤) − ✶ ♣✵ + π(❤) ❤ − q(❤) / ρ ∈ R✼[❤] ❉✐♠❡♥s✐♦♥✿ ✷ ◆✉♠❡r✐❝❛❧ s❝❤❡♠❡✿ ❋r❡❡❋❡♠✰✰ ✭❝♦♥✈❡❝t✮

❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✹ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s Pr❡s❡♥t❛t✐♦♥ ♦❢ Pr♦❥❡❝t ❉✐P▲♦▼❛

▼♦❞❡❧s ❛♥❞ ♥✉♠❡r✐❝❛❧ ❛s♣❡❝ts

▲▼◆❈ ♠♦❞❡❧ ✭✈❡rs✐♦♥ ✷✵✶✺✮            ∇· ✉ = β(❤, ♣✵) ♣✵

• Φ + ∇·
• λ(❤, ♣✵)∇❚(❤, ♣✵)
• ,

✭✶❛✮ ρ(❤, ♣✵) ×

• ∂t❤ + ✉ · ∇❤
• = Φ + ∇·
• λ(❤, ♣✵)∇❚(❤, ♣✵)
• ,

✭✶❜✮ ρ(❤, ♣✵) ×

• ∂t✉ + (✉ · ∇)✉
• − ∇· σ(✉) + ∇¯

♣ = ρ(❤, ♣✵)❣. ✭✶❝✮ ❊q✉❛t✐♦♥ ♦❢ st❛t❡✿ st✐✛❡♥❡❞ ❣❛s✴t❛❜✉❧❛t❡❞ ❧❛✇ ρ(❤, ♣✵) = γ(❤) γ(❤) − ✶ ♣✵ + π(❤) ❤ − q(❤) / ρ ∈ R✼[❤] ❉✐♠❡♥s✐♦♥✿ ✶✴✷✴✸ ◆✉♠❡r✐❝❛❧ s❝❤❡♠❡✿ ▼❖❈ ✭❋♦rtr❛♥✮✴❋r❡❡❋❡♠✰✰ ✭❝♦♥✈❡❝t✮

❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✹ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s Pr❡s❡♥t❛t✐♦♥ ♦❢ Pr♦❥❡❝t ❉✐P▲♦▼❛

▼♦❞❡❧s ❛♥❞ ♥✉♠❡r✐❝❛❧ ❛s♣❡❝ts

▲▼◆❈ ♠♦❞❡❧ ✭✈❡rs✐♦♥ ✷✵✶❄✮              ∇· ✉ = − P′

✵(t)

ρ

• ❤, P✵(t)
• ❝✷

❤, P✵(t) + β

• ❤, P✵(t)
• P✵(t)

Φ, ✭✶❛✮ ρ

• ❤, P✵(t)
• ×
• ∂t❤ + ✉ · ∇❤
• = Φ + P′

✵(t),

✭✶❜✮ ρ

• ❤, P✵(t)
• ×
• ∂t✉ + (✉ · ∇)✉
• − ∇· σ(✉) + ∇¯

♣ = ρ

• ❤, P✵(t)
• ❣.

✭✶❝✮ ❊q✉❛t✐♦♥ ♦❢ st❛t❡✿ st✐✛❡♥❡❞ ❣❛s✴t❛❜✉❧❛t❡❞ ❧❛✇ ρ

• ❤, P✵(t)
• =

γ

• ❤, P✵(t)
• γ
• ❤, P✵(t)
• − ✶

P✵(t) + π

• ❤, P✵(t)
• ❤ − q
• ❤, P✵(t)
• /

ρ ∈ R✼[❤, P✵(t)] ❉✐♠❡♥s✐♦♥✿ ✶✴✷✴✸ ◆✉♠❡r✐❝❛❧ s❝❤❡♠❡✿ ▼❖❈ ✭❋♦rtr❛♥✮✴❋r❡❡❋❡♠✰✰ ✭❝♦♥✈❡❝t✮

❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✹ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s Pr❡s❡♥t❛t✐♦♥ ♦❢ Pr♦❥❡❝t ❉✐P▲♦▼❛

❋r♦♠ P❛r✐s ✇✐t❤ ❧♦✈❡

❲❤❡♥❄ ✺✳✕✻✳ ◆♦✈❡♠❜❡r ✷✵✶✺ ❲❤❡r❡❄ ❯♥✐✈✳ P❛r✐s ❉❡s❝❛rt❡s ❆❜♦✉t❄ ▲♦✇ ▼❛❝❤ ❛♥❞ ❧♦✇ ❋r♦✉❞❡ ✢♦✇s ❆❞❞r❡ss❡❞ t♦ ✇❤♦♠❄ ❆♥②❜♦❞② ✐♥t❡r❡st❡❞ ✐♥ t❤❡♦r❡t✐❝❛❧ ❛♥❞ ♥✉♠❡r✐❝❛❧ ❛s♣❡❝ts ❆♥❞❄ P♦st❡r s❡ss✐♦♥ ✭❛♣♣❧✐❝❛t✐♦♥s ❜❡❢♦r❡ ✷✺✳ ❙❡♣t❡♠❜❡r✮

❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✺ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s ■♥tr♦❞✉❝t✐♦♥

▼❡t❤♦❞ ♦❢ ❝❤❛r❛❝t❡r✐st✐❝s

P✉r♣♦s❡✿ ❞❡s✐❣♥✐♥❣ ❛♥ ❛❝❝✉r❛t❡ ♥✉♠❡r✐❝❛❧ s❝❤❡♠❡ ❢♦r s♠♦♦t❤ ❢✉♥❝t✐♦♥s ♦❢ ❛❞✈❡❝✲ t✐♦♥ ❡q✉❛t✐♦♥s ✇❤✐❧❡ s❛t✐s❢②✐♥❣ ♣❤②s✐❝❛❧ ❝♦♥str❛✐♥ts ❛t t❤❡ ❞✐s❝r❡t❡ ❧❡✈❡❧✳ ❆❞✈❡❝t✐♦♥ ❡q✉❛t✐♦♥ ∂t❨ + ❯ · ∇❨ = ❢ . ▼❡t❤♦❞ ♦❢ ❝❤❛r❛❝t❡r✐st✐❝s ❞ ❞t

• ❨
• t, X(t; s, ①)
• = ❢
• t, X(t; s, ①)
• ,

❢♦r X s♦❧✉t✐♦♥ t♦✿      ❞X ❞t = ❯

• t, X(t; s, ①)
• ,

X(s; s, ①) = ①.

❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✻ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s ■♥tr♦❞✉❝t✐♦♥

P❤②s✐❝❛❧ ❝♦♥str❛✐♥ts

❚❤✐s ❛❧s♦ r❡❛❞s ❨(t + ∆t, ①) = ❨

• t, X(t; t + ∆t, ①)
• +

t+∆t

t

❢

• σ, X(σ; t + ∆t, ①)
• ❞σ,

t t + ∆t ① X(t; t + ∆t, ①)

❧ ■❢ ❢ = ✵✱ ❨ ✐s ❝♦♥st❛♥t ❛❧♦♥❣ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ❝✉r✈❡s✿ ♠❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡✳ ❧ ■❢ ❢ > ✵✱ ❨ ✐s ♠♦♥♦t♦♥❡✲✐♥❝r❡❛s✐♥❣ ❛❧♦♥❣ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ❝✉r✈❡s✿ ♣♦s✐t✐✈✐t② ♣r✐♥❝✐♣❧❡✳

❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✼ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s ■♥tr♦❞✉❝t✐♦♥

❆♣♣❧✐❝❛t✐♦♥s

❆❜str❛❝t ❇✉❜❜❧❡ ❱✐❜r❛t✐♦♥ ♠♦❞❡❧ ✭P✳✱ ❉■❊✱ ✷✻✭✶✲✷✮✱ ✺✾✲✽✵✱ ✷✵✶✸✮        ∂t❨ + ∇φ · ∇❨ = ✵, ∆φ = ψ(t)

• ❨ − ✶

|Ω|

• Ω

❨(t, ②) ❞②

• ❊①♣❧✐❝✐t s♦❧✉t✐♦♥ ✐♥ ✶❉ ❢♦r Ω = (✵, ✶)

❨(t, ①) = ❨✵ Θ−✶

t (①)

• ✇❤❡r❡

Θt(①) = ①

✵

❡Ψ(t)❨✵(②)❞② ✶

✵

❡Ψ(t)❨✵(②)❞②

❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✽ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s ■♥tr♦❞✉❝t✐♦♥

❆♣♣❧✐❝❛t✐♦♥s

▲♦✇ ▼❛❝❤ ◆✉❝❧❡❛r ❈♦r❡ ♠♦❞❡❧ ✭❇❡r♥❛r❞ ❡t ❛❧✱ ▼✷❆◆✱ ✹✽✭✵✻✮✱ ✶✻✸✾✲✶✻✼✾✱ ✷✵✶✹✮      ∂t❤ + ✈∂②❤ = βℓΦ(②)

♣✵

(❤ − qℓ), ∂②✈ = βℓΦ(②)

♣✵

. ❊①♣❧✐❝✐t s♦❧✉t✐♦♥ ✐♥ ✶❉ ❢♦r Ω = (✵, ✶) ❤(t, ②) = qℓ + ✈(②) ×          ❤✵

• Θ−✶(Θ(②) − t)
• − qℓ

✈ (Θ−✶(Θ(②) − t)) , ✐❢ Θ(②) ≥ t, ❤❡

• t − Θ(②)
• − qℓ

✈❡ , ♦t❤❡r✇✐s❡✱ ✇✐t❤ ✈(②) = ✈❡ + βℓ ♣✵ ②

✵

Φ(③) ❞③ ❛♥❞ Θ(②) = ②

✵

❞③ ✈(③)✳

❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✽ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s ◆✉♠❡r✐❝s

❙❡tt✐♥❣s

• ✐✈❡♥ ❛ ♠❡s❤ s✐③❡ ∆①✱ ❛ t✐♠❡ st❡♣ ∆t ❛♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ t✐♠❡✲s♣❛❝❡ ❣r✐❞

(t♥, ①✐)✱ ✇❡ ❞❡s✐❣♥ ❛ ♥✉♠❡r✐❝❛❧ s❝❤❡♠❡ ❜❛s❡❞ ♦♥ t❤❡ r❡s♦❧✉t✐♦♥ ♦❢ t❤❡ t✇♦ st❡♣s ♦❢ t❤❡ ♠❡t❤♦❞ ♦❢ ❝❤❛r❛❝t❡r✐st✐❝s ✭▼❖❈ s❝❤❡♠❡s✮✳ ❲❡ ❝♦♥s✐❞❡r ❤❡r❡ ❛ ❜❛❝❦✇❛r❞ ♠❡t❤♦❞✳ ① ❚✐♠❡ st❡♣✿ ②✐❡❧❞s ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ξ♥

✐ := X(t♥; t♥+✶, ①✐) ✇✐t❤

     ❞X ❞t = ❯

• t, X(t; t♥+✶, ①✐)
• ,

X(t♥+✶; t♥+✶, ①✐) = ①✐. ② ❙♣❛❝❡ st❡♣✿ ②✐❡❧❞s ❛♥ ✐♥t❡r♣♦❧❛t❡❞ ✈❛❧✉❡ ♦❢ ❨

• t♥, ξ♥

✐ )

• t♦ ❝♦♠♣✉t❡

❨(t♥+✶, ①✐) = ❨

• t♥, X(t♥; t♥+✶, ①✐)
• +

t♥+✶

t♥

❢

• σ, X(σ; t♥+✶, ①✐)
• ❞σ,

❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✾ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s ◆✉♠❡r✐❝s

▼❖❈✶ s❝❤❡♠❡

t♥ t♥+✶ ①✐ ①❥ ①❥−✶ ①❥+✶ ①❥+✷ ξ♥

✐

❚❛②❧♦r ❡①♣❛♥s✐♦♥✿ X(t♥; t♥+✶, ①✐) = X(t♥; t♥, ①✐) + ∆t · ∂sX(t♥; t♥, ①✐) + O(∆t✷).

❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✶✵ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s ◆✉♠❡r✐❝s

▼❖❈✶ s❝❤❡♠❡

t♥ t♥+✶ ①✐ ①❥ ①❥−✶ ①❥+✶ ①❥+✷ ˆ ξ♥

✐,✶

ξ♥

✐

❚❛②❧♦r ❡①♣❛♥s✐♦♥✿ X(t♥; t♥+✶, ①✐) = X(t♥; t♥, ①✐) + ∆t · ∂sX(t♥; t♥, ①✐) + O(∆t✷). ❯s✐♥❣ ∂sX(s; s, ①) = −∇①X(s; s, ①)❯(s, ①) = −❯(s, ①)✱ ✇❡ s❡t✿ ξ♥

✐,✶ = ①✐ − ✉(t♥, ①✐)∆t.

❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✶✵ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s ◆✉♠❡r✐❝s

▼❖❈✶ s❝❤❡♠❡

t♥ t♥+✶ ①✐ ①❥ ①❥−✶ ①❥+✶ ①❥+✷ ˆ ξ♥

✐,✶

ξ♥

✐

❙❡tt✐♥❣ θ = ①❥+✶ − ξ♥

✐,✶

∆① ✱ ✇❡ ✜♥❛❧❧② ❝♦♠♣✉t❡ ❨ ♥+✶

✐

= θ❨ ♥

❥ + (✶ − θ)❨ ♥ ❥+✶ + ∆t · ❢ (t♥+✶, ①✐).

❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✶✵ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s ◆✉♠❡r✐❝s

▼❖❈✶ s❝❤❡♠❡

t♥ t♥+✶ ①✐ ①❥ ①❥−✶ ①❥+✶ ①❥+✷ ˆ ξ♥

✐,✶

ξ♥

✐

❙❡tt✐♥❣ θ = ①❥+✶ − ξ♥

✐,✶

∆① ✱ ✇❡ ✜♥❛❧❧② ❝♦♠♣✉t❡ ❨ ♥+✶

✐

= θ❨ ♥

❥ + (✶ − θ)❨ ♥ ❥+✶ + ∆t · ❢ (t♥+✶, ①✐).

❚❤❡ s❝❤❡♠❡ ✐s ✉♥❝♦♥❞✐t✐♦♥❛❧❧② st❛❜❧❡✱ ❝♦♥s✐st❡♥t ❛t ♦r❞❡r ✶ ❛♥❞ s❛t✐s✜❡s t❤❡ ♠❛①✐✲ ♠✉♠ ♣r✐♥❝✐♣❧❡ ✭❢♦r ❢ = ✵✮✳

❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✶✵ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s ◆✉♠❡r✐❝s

▼❖❈✶ s❝❤❡♠❡

t♥ t♥+✶ ①✐ ①❥ ①❥−✶ ①❥+✶ ①❥+✷ ˆ ξ♥

✐,✶

❋♦r ❢ = ✵ ❛♥❞ ❛ ❝♦♥st❛♥t ✈❡❧♦❝✐t② ✉✵ > ✵✱ ✐❢ ❛ ❈❋▲ ❝♦♥❞✐t✐♦♥ ∆t ≤ ∆①/✉✵ ✐s ✐♠♣♦s❡❞✱ ✇❡ r❡❝♦✈❡r t❤❡ st❛♥❞❛r❞ ✉♣✇✐♥❞ s❝❤❡♠❡✳ ❥ = ✐ − ✶, ξ♥

✐,✶ = ①✐ − ✉✵∆t, θ = ✉✵

∆t ∆① , ❨ ♥+✶

✐

= ✉✵ ∆t ∆① ❨ ♥

✐−✶ +

• ✶ − ✉✵

∆t ∆①

• ❨ ♥

✐ .

❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✶✵ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s ◆✉♠❡r✐❝s

❇♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s

❉✐r✐❝❤❧❡t✿ ❨ (t, ✵) = ❨❡(t) ① t ✵ t♥+✶ ①✐ t♥ ξ♥

✐

t∗

✐

❘❡s♦❧✉t✐♦♥ ♦❢ X(t∗

✐ ; t♥+✶, ①✐) = ✵ ❜② ♠❡❛♥s ♦❢ ❛ ✜①❡❞✲♣♦✐♥t ♠❡t❤♦❞✳

❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✶✶ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s ◆✉♠❡r✐❝s

❚♦✇❛r❞s ❤✐❣❤❡r ♦r❞❡rs✿ t❤❡ ▼❖❈✷ s❝❤❡♠❡

• ♦❛❧✿ r❡❛❝❤✐♥❣ ♦r❞❡r ✷ ✇❤✐❧❡ ♣r❡s❡r✈✐♥❣ t❤❡ ♠❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡

① ✷♥❞✲♦r❞❡r ❆♣♣r♦①✐♠❛t✐♦♥ ξ♥

✐,✷ ♦❢ t❤❡ ❢♦♦t ♦❢ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ❝✉r✈❡ ξ♥ ✐

② ✷♥❞✲♦r❞❡r ✐♥t❡r♣♦❧❛t✐♦♥ ♣r♦❝❡❞✉r❡ t♦ ❝♦♠♣✉t❡ ❨(t♥, ξ♥

✐,✷)

■ss✉❡✿ ♥♦ ❧✐♥❡❛r s❝❤❡♠❡ ♦❢ ♦r❞❡r ♣ ≥ ✷ ✐s ♠♦♥♦t♦♥✐❝✐t②✲♣r❡s❡r✈✐♥❣ ✭s❡❡ ❢♦r ✐♥st❛♥❝❡ ▲❡✈❡q✉❡✱ ✶✾✾✷✮ ❚r✐❝❦✿ ✈❛r✐❛❜❧❡ st❡♥❝✐❧ t❤r♦✉❣❤ ❛ ♥♦♥❧✐♥❡❛r ❝r✐t❡r✐♦♥ t♦ ❝❤♦♦s❡ ❜❡t✇❡❡♥ t✇♦ ♣♦t❡♥t✐❛❧ ✐♥t❡r♣♦❧❛t✐♦♥ ♣r♦❝❡ss❡s

❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✶✷ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s ◆✉♠❡r✐❝s

❈♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ❢♦♦t ξ♥

✐ ❙t❡♣ ① X(t♥; t♥+✶, ①✐) = X(t♥; t♥, ①✐) + ∆t · ∂sX(t♥; t♥, ①✐) + ∆t✷ ✷ ∂ssX(t♥; t♥, ①✐) + O(∆t✸). Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ✢♦✇ ∂sX(s; s, ①) = −∇①X(s; s, ①)❯(s, ①) = −❯(s, ①), ∂✷

ssX(s; s, ①) = −∂t❯(s, ①) − ∂sX ❚(s; s, ①)∇①❯(s, ①).

❍❡♥❝❡ ξ♥

✐ = ①✐ − ✉♥ ✐ ∆t + ∆t✷

✷ [✉♥

✐ ∂①✉♥ ✐ − ∂t✉♥ ✐ ] + O(∆t✸),

ξ♥

✐,✷ = ①✐ − ∆t ✸✉♥ ✐ − ✉♥ ✐−✶

✷ + ∆t✷ ✷ ✉♥

✐ ∂①✉♥ ✐ .

❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✶✸ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s ◆✉♠❡r✐❝s

■♥t❡r♣♦❧❛t✐♦♥ ♣r♦❝❡❞✉r❡

❙t❡♣ ②

t♥ t♥+✶ ①✐ ①❥ ①❥−✶ ①❥+✶ ①❥+✷ ξ♥

✐

❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✶✹ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s ◆✉♠❡r✐❝s

■♥t❡r♣♦❧❛t✐♦♥ ♣r♦❝❡❞✉r❡

❙t❡♣ ②

t♥ t♥+✶ ①✐ ①❥ ①❥−✶ ①❥+✶ ①❥+✷ ˆ ξ♥

✐,✶

ˆ ξ♥

✐,✷

ξ♥

✐

❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✶✹ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s ◆✉♠❡r✐❝s

■♥t❡r♣♦❧❛t✐♦♥ ♣r♦❝❡❞✉r❡

❙t❡♣ ②

t♥ t♥+✶ ①✐ ①❥ ①❥−✶ ①❥+✶ ①❥+✷ Yℓ ˆ ξ♥

✐,✷

❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✶✹ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s ◆✉♠❡r✐❝s

■♥t❡r♣♦❧❛t✐♦♥ ♣r♦❝❡❞✉r❡

❙t❡♣ ②

t♥ t♥+✶ ①✐ ①❥ ①❥−✶ ①❥+✶ ①❥+✷ Yr ˆ ξ♥

✐,✷

❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✶✹ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s ◆✉♠❡r✐❝s

■♥t❡r♣♦❧❛t✐♦♥ ♣r♦❝❡❞✉r❡

❙t❡♣ ②

t♥ t♥+✶ ①✐ ①❥ ①❥−✶ ①❥+✶ ①❥+✷ ˆ ξ♥

✐,✷

❚r✐❝❦✿ ❛❞❛♣t✐✈❡ st❡♥❝✐❧

❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✶✹ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s ◆✉♠❡r✐❝s

❍♦✇ t♦ s❡❧❡❝t t❤❡ r❡❧❡✈❛♥t s❝❤❡♠❡❄

❲❡ s❡t θ♥

✐❥ =

①❥+✶ − ξ♥

✐,✷

∆① ✳ ❚❤❡ s❝❤❡♠❡ r❡❛❞s ❨ ♥+✶

✐

= α♥

✐❥

• Yℓ(θ♥

✐❥ )

• −

θ♥

✐❥(✶ − θ♥ ✐❥)

✷ ❨ ♥

❥−✶ + θ♥ ✐❥(✷ − θ♥ ✐❥)❨ ♥ ❥ +

(✶ − θ♥

✐❥)(✷ − θ♥ ✐❥)

✷ ❨ ♥

❥+✶

• + (✶ − α♥

✐❥)

θ♥

✐❥(✶ + θ♥ ✐❥)

✷ ❨ ♥

❥ +

• ✶ −
• θ♥

✐❥

✷ ❨ ♥

❥+✶ −

θ♥

✐❥(✶ − θ♥ ✐❥)

✷ ❨ ♥

❥+✷

• Yr (θ♥

✐❥ )

• ❲❡ ♠✉st s❛t✐s❢②

❨ ♥+✶

✐

∈

• ♠✐♥

❦∈V(❥) ❨ ♥ ❦ , ♠❛① ❦∈V(❥) ❨ ♥ ❦

• ,

V(❥) = {❥, ❥ + ✶}.

❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✶✺ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s ◆✉♠❡r✐❝s

❍♦✇ t♦ s❡❧❡❝t t❤❡ r❡❧❡✈❛♥t s❝❤❡♠❡❄

❳❥−✶ ❳❥ ❳❥+✶ ❨ θ ✷ ✶ ✵ κ+ κ−

❋♦r ❛ ❝♦♥st❛♥t ✈❡❧♦❝✐t② ✉✵ ❛♥❞ ✉♥❞❡r ❛ ❈❋▲ ❝♦♥❞✐t✐♦♥ ∆t ≤ ∆①/✉✵✱ t❤❡ ❧❡❢t s❝❤❡♠❡ ✐s t❤❡ ❇❡❛♠✲❲❛r♠✐♥❣ s❝❤❡♠❡ ✭✐❢ ✉✵ > ✵✮ ♦r t❤❡ ▲❛①✲❲❡♥❞r♦✛ s❝❤❡♠❡ ✭✐❢ ✉✵ < ✵✮✳

❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✶✺ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s ◆✉♠❡r✐❝s

❍♦✇ t♦ s❡❧❡❝t t❤❡ r❡❧❡✈❛♥t s❝❤❡♠❡❄

❚❤❡ ❧❡❢t s❝❤❡♠❡ ✭r❡s♣✳ t❤❡ ✏r✐❣❤t✑ s❝❤❡♠❡✮ s❛t✐s✜❡s t❤❡ ♠❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ✐✛ θ♥

✐❥ ∈ (κ− ℓ , κ+ ℓ )

(r❡s♣✳ (κ−

r , κ+ r )),

✇✐t❤ κ−

ℓ =

✷(❨ ♥

❥+✶ − ❨ ♥ ❥ )

❨ ♥

❥−✶ − ✷❨ ♥ ❥ + ❨ ♥ ❥+✶

, κ−

r =

✷(❨ ♥

❥+✶ − ❨ ♥ ❥ )

❨ ♥

❥ − ✷❨ ♥ ❥+✶ + ❨ ♥ ❥+✷

, κ+

ℓ =

❨ ♥

❥−✶ − ✹❨ ♥ ❥ + ✸❨ ♥ ❥+✶

❨ ♥

❥−✶ − ✷❨ ♥ ❥ + ❨ ♥ ❥+✶

, κ+

r =

❨ ♥

❥+✷ − ❨ ♥ ❥

❨ ♥

❥ − ✷❨ ♥ ❥+✶ + ❨ ♥ ❥+✷

. ✭P✳✱ ❉❈❉❙✲❙✱ ✺✭✸✮✱ ✻✹✶✲✻✺✻✱ ✷✵✶✷✮

❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✶✺ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s ◆✉♠❡r✐❝s

❍♦✇ t♦ s❡❧❡❝t t❤❡ r❡❧❡✈❛♥t s❝❤❡♠❡❄

■♥ ❨ ♥+✶

✐

= α♥

✐❥Yℓ(θ♥ ✐❥) + (✶ − α♥ ✐❥)Yr(θ♥ ✐❥)✱ α♥ ✐❥ ✐s ❞❡t❡r♠✐♥❡❞ ❧✐❦❡ t❤✐s

❧ ■❢ θ♥

✐❥ ∈ (κ− ℓ , κ+ ℓ ) ∪ (κ− r , κ+ r )✱ ✇❡ s❡t

α♥

✐❥ =

✶ + θ♥

✐❥

✸ ❧ ■❢ θ♥

✐❥ ∈ (κ− ℓ , κ+ ℓ ) ❜✉t θ♥ ✐❥ ∈ [κ− r , κ+ r ]✱ ✇❡ s❡t

α♥

✐❥ = ✶

❧ ■❢ θ♥

✐❥ ∈ (κ− r , κ+ r ) ❜✉t θ♥ ✐❥ ∈ [κ− ℓ , κ+ ℓ ]✱ ✇❡ s❡t

α♥

✐❥ = ✵

❧ ■❢ θ♥

✐❥ ∈ [κ− ℓ , κ+ ℓ ] ∩ [κ− r , κ+ r ]✱ ✇❡ s❡t

α♥

✐❥ = θ♥ ✐❥, Yℓ(θ♥ ✐❥) = ❨ ♥ ❥ , Yr(θ♥ ✐❥) = ❨ ♥ ❥+✶

❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✶✺ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s ◆✉♠❡r✐❝s

◆✉♠❡r✐❝❛❧ r❡s✉❧ts

❊①❛♠♣❧❡✿ ❚r❛♥s♣♦rt ♦❢ ❛ ✈♦❧✉♠❡ ❢r❛❝t✐♦♥

❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✶✻ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s ◆✉♠❡r✐❝s

◆✉♠❡r✐❝❛❧ r❡s✉❧ts

❊①❛♠♣❧❡✿ ❚r❛♥s♣♦rt ♦❢ ❛ ✈♦❧✉♠❡ ❢r❛❝t✐♦♥

❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✶✻ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s ◆✉♠❡r✐❝s

◆✉♠❡r✐❝❛❧ r❡s✉❧ts

❊①❛♠♣❧❡✿ ❚r❛♥s♣♦rt ✇✐t❤ s♦✉r❝❡ t❡r♠

❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✶✻ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s ◆✉♠❡r✐❝s

◆✉♠❡r✐❝❛❧ r❡s✉❧ts

❊①❛♠♣❧❡✿ ❚r❛♥s♣♦rt ✇✐t❤ s♦✉r❝❡ t❡r♠

❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✶✻ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s ◆✉♠❡r✐❝s

◆✉♠❡r✐❝❛❧ r❡s✉❧ts

❊①❛♠♣❧❡✿ ❚r❛♥s♣♦rt ✇✐t❤ s♦✉r❝❡ t❡r♠

❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✶✻ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s ◆✉♠❡r✐❝s

◆✉♠❡r✐❝❛❧ r❡s✉❧ts

❊①❛♠♣❧❡✿ ❚r❛♥s♣♦rt ✇✐t❤ s♦✉r❝❡ t❡r♠

❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✶✻ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s ◆✉♠❡r✐❝s

◆✉♠❡r✐❝❛❧ r❡s✉❧ts

❊①❛♠♣❧❡✿ ❚r❛♥s♣♦rt ✇✐t❤ s♦✉r❝❡ t❡r♠

❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✶✻ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s ◆✉♠❡r✐❝s

◆✉♠❡r✐❝❛❧ r❡s✉❧ts

❊①❛♠♣❧❡✿ ❚r❛♥s♣♦rt ✇✐t❤ s♦✉r❝❡ t❡r♠

❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✶✻ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s ◆✉♠❡r✐❝s

◆✉♠❡r✐❝❛❧ r❡s✉❧ts

❊①❛♠♣❧❡✿ ❚r❛♥s♣♦rt ✇✐t❤ s♦✉r❝❡ t❡r♠

❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✶✻ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s ◆✉♠❡r✐❝s

◆✉♠❡r✐❝❛❧ r❡s✉❧ts

❊①❛♠♣❧❡✿ ❚r❛♥s♣♦rt ✇✐t❤ s♦✉r❝❡ t❡r♠

❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✶✻ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s ◆✉♠❡r✐❝s

◆✉♠❡r✐❝❛❧ r❡s✉❧ts

❊①❛♠♣❧❡✿ ❚r❛♥s♣♦rt ✇✐t❤ s♦✉r❝❡ t❡r♠

❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✶✻ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s ◆✉♠❡r✐❝s

◆✉♠❡r✐❝❛❧ r❡s✉❧ts

❊①❛♠♣❧❡✿ ❚r❛♥s♣♦rt ✇✐t❤ s♦✉r❝❡ t❡r♠

❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✶✻ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s ◆✉♠❡r✐❝s

◆✉♠❡r✐❝❛❧ r❡s✉❧ts

❊①❛♠♣❧❡✿ ❚r❛♥s♣♦rt ✇✐t❤ s♦✉r❝❡ t❡r♠

❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✶✻ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s ◆✉♠❡r✐❝s

◆✉♠❡r✐❝❛❧ r❡s✉❧ts

❊①❛♠♣❧❡✿ ❚r❛♥s♣♦rt ✇✐t❤ s♦✉r❝❡ t❡r♠

❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✶✻ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s ❊①t❡♥s✐♦♥s ♦❢ t❤❡ ▼❖❈ s❝❤❡♠❡s

❚❤❡ ■◆❚▼❖❈ ✈❡rs✐♦♥

■♥ t❤❡ ▲▼◆❈✲❙● ♠♦❞❡❧✱ ✇❡ ❤❛✈❡ t♦ ❞❡❛❧ ✇✐t❤ ρ(❤, ♣✵)

• ∂t❤ + ✈∂②❤
• = Φ,

ρ(❤, ♣✵) = ♣✵ β(❤) ·

• ❤ − q(❤)
• ❆s β ❛♥❞ q ❛r❡ ♣✐❡❝❡✇✐s❡ ❝♦♥st❛♥t✱ ✐t ✐s ❡❛s② t♦ ❞❡t❡r♠✐♥❡ t❤❡ ♣r✐♠✐t✐✈❡ ❢✉♥❝t✐♦♥ ❘

♦❢ ❤ → ρ(❤, ♣✵) ❛s ✇❡❧❧ ❛s ✐ts ✐♥✈❡rs❡ ❘−✶✳

❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✶✼ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s ❊①t❡♥s✐♦♥s ♦❢ t❤❡ ▼❖❈ s❝❤❡♠❡s

❚❤❡ ■◆❚▼❖❈ ✈❡rs✐♦♥

■♥ t❤❡ ▲▼◆❈✲❙● ♠♦❞❡❧✱ ✇❡ ❤❛✈❡ t♦ ❞❡❛❧ ✇✐t❤ ρ(❤, ♣✵)

• ∂t❤ + ✈∂②❤
• = Φ,

ρ(❤, ♣✵) = ♣✵ β(❤) ·

• ❤ − q(❤)
• ❆s β ❛♥❞ q ❛r❡ ♣✐❡❝❡✇✐s❡ ❝♦♥st❛♥t✱ ✐t ✐s ❡❛s② t♦ ❞❡t❡r♠✐♥❡ t❤❡ ♣r✐♠✐t✐✈❡ ❢✉♥❝t✐♦♥ ❘

♦❢ ❤ → ρ(❤, ♣✵) ❛s ✇❡❧❧ ❛s ✐ts ✐♥✈❡rs❡ ❘−✶✳

❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✶✼ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s ❊①t❡♥s✐♦♥s ♦❢ t❤❡ ▼❖❈ s❝❤❡♠❡s

❉✐✛✉s✐♦♥

■♥ t❤❡ λ✲▲▼◆❈ ♠♦❞❡❧✱ ✇❡ ❤❛✈❡ t♦ ❞❡❛❧ ✇✐t❤ ∂t❤ + ✈∂②❤ = Φ + λ∂✷

②②❤

ρ(❤, ♣✵) ❋✐rst ❛tt❡♠♣t ❤♥+✶

✐

= ❤(t♥, ξ♥

✐ ) + ∆t

Φ(t♥, ξ♥

✐ )

ρ

• ❤(t♥, ξ♥

✐ ), ♣✵

+ λ∆t ρ

• ❤(t♥, ξ♥

✐ ), ♣✵

• ∂✷

②②❤(t♥, ξ♥ ✐ )

• ≈θ∂✷

②② ❤♥ ❥ +(✶−θ)∂✷ ②② ❤♥ ❥+✶

❙t❛❜✐❧✐t② ❝♦♥❞✐t✐♦♥✿ ∆t ∆② ✷

❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✶✽ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s ❊①t❡♥s✐♦♥s ♦❢ t❤❡ ▼❖❈ s❝❤❡♠❡s

❉✐✛✉s✐♦♥

■♥ t❤❡ λ✲▲▼◆❈ ♠♦❞❡❧✱ ✇❡ ❤❛✈❡ t♦ ❞❡❛❧ ✇✐t❤ ∂t❤ + ✈∂②❤ = Φ + λ∂✷

②②❤

ρ(❤, ♣✵) ❋✐rst ❛tt❡♠♣t ❤♥+✶

✐

= ❤(t♥, ξ♥

✐ ) + ∆t

Φ(t♥, ξ♥

✐ )

ρ

• ❤(t♥, ξ♥

✐ ), ♣✵

+ λ∆t ρ

• ❤(t♥, ξ♥

✐ ), ♣✵

• ∂✷

②②❤(t♥, ξ♥ ✐ )

• ≈θ∂✷

②② ❤♥ ❥ +(✶−θ)∂✷ ②② ❤♥ ❥+✶

❙t❛❜✐❧✐t② ❝♦♥❞✐t✐♦♥✿ ∆t ∆② ✷

❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✶✽ ✴ ✶✾ / / :

❉❡s✐❣♥ ♦❢ ▼❖❈ s❝❤❡♠❡s ❊①t❡♥s✐♦♥s ♦❢ t❤❡ ▼❖❈ s❝❤❡♠❡s

❉✐✛✉s✐♦♥

■♥ t❤❡ λ✲▲▼◆❈ ♠♦❞❡❧✱ ✇❡ ❤❛✈❡ t♦ ❞❡❛❧ ✇✐t❤ ∂t❤ + ✈∂②❤ = Φ + λ∂✷

②②❤

ρ(❤, ♣✵) ❙❡❝♦♥❞ ❛tt❡♠♣t ❤♥+✶

✐

= ❤(t♥, ξ♥

✐ ) + ∆t

Φ(t♥, ξ♥

✐ )

ρ

• ❤(t♥, ξ♥

✐ ), ♣✵

+ λ∆t ρ

• ❤(t♥, ξ♥

✐ ), ♣✵

∂✷

②②❤♥+✶ ✐

❚❤❡ r❡s✉❧t✐♥❣ ❧✐♥❡❛r s②st❡♠ ✐♥✈♦❧✈❡s ❛ tr✐❞✐❛❣♦♥❛❧ ▼✲♠❛tr✐①✳ ❚❤❡ ❚❤♦♠❛s ❛❧❣♦r✐t❤♠ ❡♥❛❜❧❡s t♦ ✐♥✈❡rt ✐♥ ❛ O(◆②) ♣r♦❝❡❞✉r❡✳

❨✳ P❡♥❡❧ ✕ ❈❉▼❆❚❍ ✲ ✶✽ ✴ ✶✾ / / :

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥