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Oct 14, 2022 235 likes •935 views

rr s r t str qts t tr

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SLIDE 1

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥

❈✳ ❇❡rt❤♦♥✶✱ ❙✳ ❈❧❛✐♥✷✱ ❋✳ ❋♦✉❝❤❡r✶✱✸✱ ❘✳ ▲♦✉❜èr❡✹✱ ❱✳ ▼✐❝❤❡❧✲❉❛♥s❛❝✺

1▲❛❜♦r❛t♦✐r❡ ❞❡ ▼❛t❤é♠❛t✐q✉❡s ❏❡❛♥ ▲❡r❛②✱ ❯♥✐✈❡rs✐té ❞❡ ◆❛♥t❡s 2❈❡♥tr❡ ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ▼✐♥❤♦ ❯♥✐✈❡rs✐t② 3➱❝♦❧❡ ❈❡♥tr❛❧❡ ❞❡ ◆❛♥t❡s 4❈◆❘❙ ❡t ■♥st✐t✉t ❞❡ ▼❛t❤é♠❛t✐q✉❡s ❞❡ ❇♦r❞❡❛✉① 5■♥st✐t✉t ❞❡ ▼❛t❤é♠❛t✐q✉❡s ❞❡ ❚♦✉❧♦✉s❡ ❡t ■◆❙❆ ❚♦✉❧♦✉s❡

❚✉❡s❞❛②✱ ❋❡❜r✉❛r② ✷✼t❤✱ ✷✵✶✽ ❙é♠✐♥❛✐r❡ ❆❈❙■❖▼✱ ▼♦♥t♣❡❧❧✐❡r

slide-2
SLIDE 2

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ■♥tr♦❞✉❝t✐♦♥ ❛♥❞ ♠♦t✐✈❛t✐♦♥s

❙❡✈❡r❛❧ ❦✐♥❞s ♦❢ ❞❡str✉❝t✐✈❡ ❣❡♦♣❤②s✐❝❛❧ ✢♦✇s

❉❛♠ ❢❛✐❧✉r❡ ✭▼❛❧♣❛ss❡t✱ ❋r❛♥❝❡✱ ✶✾✺✾✮ ❚s✉♥❛♠✐ ✭❚✠ ♦❤♦❦✉✱ ❏❛♣❛♥✱ ✷✵✶✶✮ ❋❧♦♦❞ ✭▲❛ ❋❛✉t❡ s✉r ▼❡r✱ ❋r❛♥❝❡✱ ✷✵✶✵✮ ▼✉❞s❧✐❞❡ ✭▼❛❞❡✐r❛✱ P♦rt✉❣❛❧✱ ✷✵✶✵✮

✶ ✴ ✹✶

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SLIDE 3

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ■♥tr♦❞✉❝t✐♦♥ ❛♥❞ ♠♦t✐✈❛t✐♦♥s

❚❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ❛♥❞ t❤❡✐r s♦✉r❝❡ t❡r♠s

     ∂th + ∂x(hu) = 0 ∂t(hu) + ∂x

  • hu2 + 1

2gh2

  • = −gh∂xZ − kq|q|

h7

  • 3

✭✇✐t❤ q = hu) ❲❡ ❝❛♥ r❡✇r✐t❡ t❤❡ ❡q✉❛t✐♦♥s ❛s ∂tW + ∂xF(W) = S(W)✱ ✇✐t❤ W = h q

  • ✳

x h(x, t)

water surface channel bottom

u(x, t) Z(x)

Z(x) ✐s t❤❡ ❦♥♦✇♥ t♦♣♦❣r❛♣❤② k ✐s t❤❡ ▼❛♥♥✐♥❣ ❝♦❡✣❝✐❡♥t g ✐s t❤❡ ❣r❛✈✐t❛t✐♦♥❛❧ ❝♦♥st❛♥t ✇❡ ❧❛❜❡❧ t❤❡ ✇❛t❡r ❞✐s❝❤❛r❣❡ q := hu

✷ ✴ ✹✶

slide-4
SLIDE 4

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ■♥tr♦❞✉❝t✐♦♥ ❛♥❞ ♠♦t✐✈❛t✐♦♥s

❙t❡❛❞② st❛t❡ s♦❧✉t✐♦♥s

❉❡✜♥✐t✐♦♥✿ ❙t❡❛❞② st❛t❡ s♦❧✉t✐♦♥s W ✐s ❛ st❡❛❞② st❛t❡ s♦❧✉t✐♦♥ ✐✛ ∂tW = 0✱ ✐✳❡✳ ∂xF(W) = S(W)✳ ❚❛❦✐♥❣ ∂tW = 0 ✐♥ t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ❧❡❛❞s t♦      ∂xq = 0 ∂x q2 h + 1 2gh2

  • = −gh∂xZ − kq|q|

h7

  • 3

. ❚❤❡ st❡❛❞② st❛t❡ s♦❧✉t✐♦♥s ❛r❡ t❤❡r❡❢♦r❡ ❣✐✈❡♥ ❜②      q = cst = q0 ∂x q2 h + 1 2gh2

  • = −gh∂xZ − kq0|q0|

h7

  • 3

.

✸ ✴ ✹✶

slide-5
SLIDE 5

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ■♥tr♦❞✉❝t✐♦♥ ❛♥❞ ♠♦t✐✈❛t✐♦♥s

❆ r❡❛❧✲❧✐❢❡ s✐♠✉❧❛t✐♦♥✿ t❤❡ ✷✵✶✶ ❚✠ ♦❤♦❦✉ ts✉♥❛♠✐✳ ❚❤❡ ✇❛t❡r ✐s ❝❧♦s❡ t♦ ❛ st❡❛❞② st❛t❡ ❛t r❡st ❢❛r ❢r♦♠ t❤❡ ts✉♥❛♠✐✳ ❚❤✐s st❡❛❞② st❛t❡ ✐s ♥♦t ♣r❡s❡r✈❡❞ ❜② ❛ ♥♦♥✲✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡✦

✹ ✴ ✹✶

slide-6
SLIDE 6

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ■♥tr♦❞✉❝t✐♦♥ ❛♥❞ ♠♦t✐✈❛t✐♦♥s

❆ r❡❛❧✲❧✐❢❡ s✐♠✉❧❛t✐♦♥✿ t❤❡ ✷✵✶✶ ❚✠ ♦❤♦❦✉ ts✉♥❛♠✐✳ ❚❤❡ ✇❛t❡r ✐s ❝❧♦s❡ t♦ ❛ st❡❛❞② st❛t❡ ❛t r❡st ❢❛r ❢r♦♠ t❤❡ ts✉♥❛♠✐✳ ❚❤✐s st❡❛❞② st❛t❡ ✐s ♥♦t ♣r❡s❡r✈❡❞ ❜② ❛ ♥♦♥✲✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡✦

✹ ✴ ✹✶

slide-7
SLIDE 7

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ■♥tr♦❞✉❝t✐♦♥ ❛♥❞ ♠♦t✐✈❛t✐♦♥s

❖❜❥❡❝t✐✈❡s

❖✉r ❣♦❛❧ ✐s t♦ ❞❡r✐✈❡ ❛ ♥✉♠❡r✐❝❛❧ ♠❡t❤♦❞ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ♠♦❞❡❧ ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ t❤❛t ❡①❛❝t❧② ♣r❡s❡r✈❡s ✐ts st❛t✐♦♥❛r② s♦❧✉t✐♦♥s ♦♥ ❡✈❡r② ♠❡s❤✳ ❚♦ t❤❛t ❡♥❞✱ ✇❡ s❡❡❦ ❛ ♥✉♠❡r✐❝❛❧ s❝❤❡♠❡ t❤❛t✿

✶ ✐s ✇❡❧❧✲❜❛❧❛♥❝❡❞ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤

t♦♣♦❣r❛♣❤② ❛♥❞ ❢r✐❝t✐♦♥✱ ✐✳❡✳ ✐t ❡①❛❝t❧② ♣r❡s❡r✈❡s ❛♥❞ ❝❛♣t✉r❡s t❤❡ st❡❛❞② st❛t❡s ✇✐t❤♦✉t ❤❛✈✐♥❣ t♦ s♦❧✈❡ t❤❡ ❣♦✈❡r♥✐♥❣ ♥♦♥❧✐♥❡❛r ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥❀

✷ ♣r❡s❡r✈❡s t❤❡ ♥♦♥✲♥❡❣❛t✐✈✐t② ♦❢ t❤❡ ✇❛t❡r ❤❡✐❣❤t❀ ✸ ❡♥s✉r❡s ❛ ❞✐s❝r❡t❡ ❡♥tr♦♣② ✐♥❡q✉❛❧✐t②❀ ✹ ❝❛♥ ❜❡ ❡❛s✐❧② ❡①t❡♥❞❡❞ ❢♦r ♦t❤❡r s♦✉r❝❡ t❡r♠s ♦❢ t❤❡

s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✭❡✳❣✳ ❜r❡❛❞t❤✮✳

✺ ✴ ✹✶

slide-8
SLIDE 8

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ■♥tr♦❞✉❝t✐♦♥ t♦ ●♦❞✉♥♦✈✲t②♣❡ s❝❤❡♠❡s

✶ ■♥tr♦❞✉❝t✐♦♥ t♦ ●♦❞✉♥♦✈✲t②♣❡ s❝❤❡♠❡s ✷ ❉❡r✐✈❛t✐♦♥ ♦❢ ❛ ✶❉ ✜rst✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ✸ ❚✇♦✲❞✐♠❡♥s✐♦♥❛❧ ❛♥❞ ❤✐❣❤✲♦r❞❡r ❡①t❡♥s✐♦♥s ✹ ✷❉ ❛♥❞ ❤✐❣❤✲♦r❞❡r ♥✉♠❡r✐❝❛❧ s✐♠✉❧❛t✐♦♥s ✺ ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ♣❡rs♣❡❝t✐✈❡s

slide-9
SLIDE 9

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ■♥tr♦❞✉❝t✐♦♥ t♦ ●♦❞✉♥♦✈✲t②♣❡ s❝❤❡♠❡s

❙❡tt✐♥❣✿ ✜♥✐t❡ ✈♦❧✉♠❡ s❝❤❡♠❡s

❖❜❥❡❝t✐✈❡✿ ❆♣♣r♦①✐♠❛t❡ t❤❡ s♦❧✉t✐♦♥ W(x, t) ♦❢ t❤❡ s②st❡♠ ∂tW + ∂xF(W) = S(W)✱ ✇✐t❤ s✉✐t❛❜❧❡ ✐♥✐t✐❛❧ ❛♥❞ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✳ ❲❡ ♣❛rt✐t✐♦♥ t❤❡ s♣❛❝❡ ❞♦♠❛✐♥ ✐♥ ❝❡❧❧s✱ ♦❢ ✈♦❧✉♠❡ ∆x ❛♥❞ ♦❢ ❡✈❡♥❧② s♣❛❝❡❞ ❝❡♥t❡rs xi✱ ❛♥❞ ✇❡ ❞❡✜♥❡✿ xi− 1

2 ❛♥❞ xi+ 1 2 ✱ t❤❡ ❜♦✉♥❞❛r✐❡s ♦❢ t❤❡ ❝❡❧❧ i❀

W n

i ✱ ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ W(x, t)✱ ❝♦♥st❛♥t ✐♥ t❤❡ ❝❡❧❧ i ❛♥❞

❛t t✐♠❡ tn✱ ✇❤✐❝❤ ✐s ❞❡✜♥❡❞ ❛s W n

i =

1 ∆x ∆x/2

∆x/2

W(x, tn)dx✳

W(x, t)

xi− 1

2

xi+ 1

2

W n

i

x x

xi xi+1 xi−1

✻ ✴ ✹✶

slide-10
SLIDE 10

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ■♥tr♦❞✉❝t✐♦♥ t♦ ●♦❞✉♥♦✈✲t②♣❡ s❝❤❡♠❡s

  • ♦❞✉♥♦✈✲t②♣❡ s❝❤❡♠❡ ✭❛♣♣r♦①✐♠❛t❡ ❘✐❡♠❛♥♥ s♦❧✈❡r✮

❆s ❛ ❝♦♥s❡q✉❡♥❝❡✱ ❛t t✐♠❡ tn✱ ✇❡ ❤❛✈❡ ❛ s✉❝❝❡ss✐♦♥ ♦❢ ❘✐❡♠❛♥♥ ♣r♦❜❧❡♠s ✭❈❛✉❝❤② ♣r♦❜❧❡♠s ✇✐t❤ ❞✐s❝♦♥t✐♥✉♦✉s ✐♥✐t✐❛❧ ❞❛t❛✮ ❛t t❤❡ ✐♥t❡r❢❛❝❡s ❜❡t✇❡❡♥ ❝❡❧❧s✿      ∂tW + ∂xF(W) = S(W) W(x, tn) =

  • W n

i ✐❢ x < xi+ 1

2

W n

i+1 ✐❢ x > xi+ 1

2

xi xi+1 xi+1

2

W n

i

W n

i+1

❋♦r S(W) = 0✱ t❤❡ ❡①❛❝t s♦❧✉t✐♦♥ t♦ t❤❡s❡ ❘✐❡♠❛♥♥ ♣r♦❜❧❡♠s ✐s ✉♥❦♥♦✇♥ ♦r ❝♦st❧② t♦ ❝♦♠♣✉t❡ ✇❡ r❡q✉✐r❡ ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥✳

✼ ✴ ✹✶

slide-11
SLIDE 11

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ■♥tr♦❞✉❝t✐♦♥ t♦ ●♦❞✉♥♦✈✲t②♣❡ s❝❤❡♠❡s

  • ♦❞✉♥♦✈✲t②♣❡ s❝❤❡♠❡ ✭❛♣♣r♦①✐♠❛t❡ ❘✐❡♠❛♥♥ s♦❧✈❡r✮

❲❡ ❝❤♦♦s❡ t♦ ✉s❡ ❛♥ ❛♣♣r♦①✐♠❛t❡ ❘✐❡♠❛♥♥ s♦❧✈❡r✱ ❛s ❢♦❧❧♦✇s✳ W n

i

W n

i+1

W n

i+ 1

2

λL

i+ 1

2

λR

i+ 1

2

xi+ 1

2

x t W n

i+ 1

2 ✐s ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ W n

i ❛♥❞

W n

i+1 ✭✐✳❡✳ ♦❢ t❤❡ s♦❧✉t✐♦♥ t♦ t❤❡ ❘✐❡♠❛♥♥ ♣r♦❜❧❡♠✮✱ ♣♦ss✐❜❧② ♠❛❞❡

♦❢ s❡✈❡r❛❧ ❝♦♥st❛♥t st❛t❡s s❡♣❛r❛t❡❞ ❜② ❞✐s❝♦♥t✐♥✉✐t✐❡s✳ λL

i+ 1

2 ❛♥❞ λR

i+ 1

2 ❛r❡ ❛♣♣r♦①✐♠❛t✐♦♥s ♦❢ t❤❡ ❧❛r❣❡st ✇❛✈❡ s♣❡❡❞s ♦❢

t❤❡ s②st❡♠✳

✽ ✴ ✹✶

slide-12
SLIDE 12

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ■♥tr♦❞✉❝t✐♦♥ t♦ ●♦❞✉♥♦✈✲t②♣❡ s❝❤❡♠❡s

  • ♦❞✉♥♦✈✲t②♣❡ s❝❤❡♠❡ ✭❛♣♣r♦①✐♠❛t❡ ❘✐❡♠❛♥♥ s♦❧✈❡r✮

x t tn+1 tn xi xi− 1

2

xi+ 1

2

W n

i

W n

i− 1

2

W n

i+ 1

2

λR

i− 1

2

λL

i+ 1

2

  • W ∆(x, tn+1)

W n

i−1

W n

i+1

❲❡ ❞❡✜♥❡ t❤❡ t✐♠❡ ✉♣❞❛t❡ ❛s ❢♦❧❧♦✇s✿ W n+1

i

:= 1 ∆x xi+ 1

2

xi− 1

2

W ∆(x, tn+1)dx. ❙✐♥❝❡ W n

i− 1

2 ❛♥❞ W n

i+ 1

2 ❛r❡ ♠❛❞❡ ♦❢ ❝♦♥st❛♥t st❛t❡s✱ t❤❡ ❛❜♦✈❡

✐♥t❡❣r❛❧ ✐s ❡❛s② t♦ ❝♦♠♣✉t❡✳

✾ ✴ ✹✶

slide-13
SLIDE 13

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ❉❡r✐✈❛t✐♦♥ ♦❢ ❛ ✶❉ ✜rst✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡

✶ ■♥tr♦❞✉❝t✐♦♥ t♦ ●♦❞✉♥♦✈✲t②♣❡ s❝❤❡♠❡s ✷ ❉❡r✐✈❛t✐♦♥ ♦❢ ❛ ✶❉ ✜rst✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ✸ ❚✇♦✲❞✐♠❡♥s✐♦♥❛❧ ❛♥❞ ❤✐❣❤✲♦r❞❡r ❡①t❡♥s✐♦♥s ✹ ✷❉ ❛♥❞ ❤✐❣❤✲♦r❞❡r ♥✉♠❡r✐❝❛❧ s✐♠✉❧❛t✐♦♥s ✺ ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ♣❡rs♣❡❝t✐✈❡s

slide-14
SLIDE 14

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ❉❡r✐✈❛t✐♦♥ ♦❢ ❛ ✶❉ ✜rst✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡

❚❤❡ ❍▲▲ ❛♣♣r♦①✐♠❛t❡ ❘✐❡♠❛♥♥ s♦❧✈❡r

❚♦ ❛♣♣r♦①✐♠❛t❡ s♦❧✉t✐♦♥s ♦❢ ∂tW + ∂xF(W) = 0✱ t❤❡ ❍▲▲ ❛♣♣r♦①✐♠❛t❡ ❘✐❡♠❛♥♥ s♦❧✈❡r ✭❍❛rt❡♥✱ ▲❛①✱ ✈❛♥ ▲❡❡r ✭✶✾✽✸✮✮ ♠❛② ❜❡ ❝❤♦s❡♥❀ ✐t ✐s ❞❡♥♦t❡❞ ❜② W ∆ ❛♥❞ ❞✐s♣❧❛②❡❞ ♦♥ t❤❡ r✐❣❤t✳

W HLL WL WR λL x t λR −∆x/2 ∆x/2

❚❤❡ ❝♦♥s✐st❡♥❝② ❝♦♥❞✐t✐♦♥ ✭❛s ♣❡r ❍❛rt❡♥ ❛♥❞ ▲❛①✮ ❤♦❧❞s ✐❢✿ 1 ∆x ∆x/2

−∆x/2

W ∆(∆t, x; WL, WR)dx = 1 ∆x ∆x/2

−∆x/2

WR(∆t, x; WL, WR)dx, ✇❤✐❝❤ ❣✐✈❡s WHLL = λRWR − λLWL λR − λL − F(WR) − F(WL) λR − λL = hHLL qHLL

  • ✳

◆♦t❡ t❤❛t✱ ✐❢ hL > 0 ❛♥❞ hR > 0✱ t❤❡♥ hHLL > 0 ❢♦r |λL| ❛♥❞ |λR| ❧❛r❣❡ ❡♥♦✉❣❤✳

✶✵ ✴ ✹✶

slide-15
SLIDE 15

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ❉❡r✐✈❛t✐♦♥ ♦❢ ❛ ✶❉ ✜rst✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡

▼♦❞✐✜❝❛t✐♦♥ ♦❢ t❤❡ ❍▲▲ ❛♣♣r♦①✐♠❛t❡ ❘✐❡♠❛♥♥ s♦❧✈❡r

❚❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t❤❡ t♦♣♦❣r❛♣❤② ❛♥❞ ❢r✐❝t✐♦♥ s♦✉r❝❡ t❡r♠s r❡❛❞ ❛s ❢♦❧❧♦✇s✿      ∂th + ∂xq = 0, ∂tq + ∂x q2 h + 1 2gh2

  • + gh∂xZ + kq|q|

h7

  • 3

= 0.

✶✶ ✴ ✹✶

slide-16
SLIDE 16

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ❉❡r✐✈❛t✐♦♥ ♦❢ ❛ ✶❉ ✜rst✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡

▼♦❞✐✜❝❛t✐♦♥ ♦❢ t❤❡ ❍▲▲ ❛♣♣r♦①✐♠❛t❡ ❘✐❡♠❛♥♥ s♦❧✈❡r

❲✐t❤ Y (t, x) := x✱ ✇❡ ❝❛♥ ❛❞❞ t❤❡ ❡q✉❛t✐♦♥s ∂tZ = 0 ❛♥❞ ∂tY = 0✱ ✇❤✐❝❤ ❝♦rr❡s♣♦♥❞ t♦ t❤❡ ✜①❡❞ ❣❡♦♠❡tr② ♦❢ t❤❡ ♣r♦❜❧❡♠✿                ∂th + ∂xq = 0, ∂tq + ∂x q2 h + 1 2gh2

  • + gh∂xZ + kq|q|

h7

  • 3

∂xY = 0, ∂tY = 0, ∂tZ = 0.

✶✶ ✴ ✹✶

slide-17
SLIDE 17

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ❉❡r✐✈❛t✐♦♥ ♦❢ ❛ ✶❉ ✜rst✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡

▼♦❞✐✜❝❛t✐♦♥ ♦❢ t❤❡ ❍▲▲ ❛♣♣r♦①✐♠❛t❡ ❘✐❡♠❛♥♥ s♦❧✈❡r

❲✐t❤ Y (t, x) := x✱ ✇❡ ❝❛♥ ❛❞❞ t❤❡ ❡q✉❛t✐♦♥s ∂tZ = 0 ❛♥❞ ∂tY = 0✱ ✇❤✐❝❤ ❝♦rr❡s♣♦♥❞ t♦ t❤❡ ✜①❡❞ ❣❡♦♠❡tr② ♦❢ t❤❡ ♣r♦❜❧❡♠✿                ∂th + ∂xq = 0, ∂tq + ∂x q2 h + 1 2gh2

  • + gh∂xZ + kq|q|

h7

  • 3

∂xY = 0, ∂tY = 0, ∂tZ = 0. ❚❤❡ ❡q✉❛t✐♦♥s ∂tY = 0 ❛♥❞ ∂tZ = 0 ✐♥❞✉❝❡ st❛t✐♦♥❛r② ✇❛✈❡s ❛ss♦❝✐❛t❡❞ t♦ t❤❡ s♦✉r❝❡ t❡r♠ ✭♦❢ ✇❤✐❝❤ q ✐s ❛ ❘✐❡♠❛♥♥ ✐♥✈❛r✐❛♥t✮✳ ❚♦ ❛♣♣r♦①✐♠❛t❡ s♦❧✉t✐♦♥s ♦❢ ∂tW + ∂xF(W) = S(W)✱ ✇❡ t❤✉s ✉s❡ t❤❡ ❛♣♣r♦①✐♠❛t❡ ❘✐❡♠❛♥♥ s♦❧✈❡r ❞✐s♣❧❛②❡❞ ♦♥ t❤❡ r✐❣❤t ✭❛ss✉♠✐♥❣ λL < 0 < λR✮✳

WL WR λL λR W ∗

L

W ∗

R

✶✶ ✴ ✹✶

slide-18
SLIDE 18

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ❉❡r✐✈❛t✐♦♥ ♦❢ ❛ ✶❉ ✜rst✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡

▼♦❞✐✜❝❛t✐♦♥ ♦❢ t❤❡ ❍▲▲ ❛♣♣r♦①✐♠❛t❡ ❘✐❡♠❛♥♥ s♦❧✈❡r

❲❡ ❤❛✈❡ ✹ ✉♥❦♥♦✇♥s t♦ ❞❡t❡r♠✐♥❡✿ W ∗

L =

h∗

L

q∗

L

  • ❛♥❞ W ∗

R =

h∗

R

q∗

R

  • ✳

WL WR λL λR W ∗

L

W ∗

R

✐s ❛ ✲❘✐❡♠❛♥♥ ✐♥✈❛r✐❛♥t ✇❡ t❛❦❡ ✭r❡❧❛t✐♦♥ ✶✮ ❚❤❡ ❍❛rt❡♥✲▲❛① ❝♦♥s✐st❡♥❝② ❣✐✈❡s ✉s t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ r❡❧❛t✐♦♥s✿ ✭r❡❧❛t✐♦♥ ✷✮✱ ✭r❡❧❛t✐♦♥ ✸✮✱ ✇❤❡r❡ ✳ ♥❡①t st❡♣✿ ♦❜t❛✐♥ ❛ ❢♦✉rt❤ r❡❧❛t✐♦♥

✶✷ ✴ ✹✶

slide-19
SLIDE 19

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ❉❡r✐✈❛t✐♦♥ ♦❢ ❛ ✶❉ ✜rst✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡

▼♦❞✐✜❝❛t✐♦♥ ♦❢ t❤❡ ❍▲▲ ❛♣♣r♦①✐♠❛t❡ ❘✐❡♠❛♥♥ s♦❧✈❡r

❲❡ ❤❛✈❡ ✹ ✉♥❦♥♦✇♥s t♦ ❞❡t❡r♠✐♥❡✿ W ∗

L =

h∗

L

q∗

L

  • ❛♥❞ W ∗

R =

h∗

R

q∗

R

  • ✳

q ✐s ❛ 0✲❘✐❡♠❛♥♥ ✐♥✈❛r✐❛♥t ✇❡ t❛❦❡ q∗

L = q∗ R = q∗ ✭r❡❧❛t✐♦♥ ✶✮

❚❤❡ ❍❛rt❡♥✲▲❛① ❝♦♥s✐st❡♥❝② ❣✐✈❡s ✉s t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ r❡❧❛t✐♦♥s✿ ✭r❡❧❛t✐♦♥ ✷✮✱ ✭r❡❧❛t✐♦♥ ✸✮✱ ✇❤❡r❡ ✳ ♥❡①t st❡♣✿ ♦❜t❛✐♥ ❛ ❢♦✉rt❤ r❡❧❛t✐♦♥

✶✷ ✴ ✹✶

slide-20
SLIDE 20

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ❉❡r✐✈❛t✐♦♥ ♦❢ ❛ ✶❉ ✜rst✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡

▼♦❞✐✜❝❛t✐♦♥ ♦❢ t❤❡ ❍▲▲ ❛♣♣r♦①✐♠❛t❡ ❘✐❡♠❛♥♥ s♦❧✈❡r

❲❡ ❤❛✈❡ ✹ ✉♥❦♥♦✇♥s t♦ ❞❡t❡r♠✐♥❡✿ W ∗

L =

h∗

L

q∗

L

  • ❛♥❞ W ∗

R =

h∗

R

q∗

R

  • ✳

q ✐s ❛ 0✲❘✐❡♠❛♥♥ ✐♥✈❛r✐❛♥t ✇❡ t❛❦❡ q∗

L = q∗ R = q∗ ✭r❡❧❛t✐♦♥ ✶✮

❚❤❡ ❍❛rt❡♥✲▲❛① ❝♦♥s✐st❡♥❝② ❣✐✈❡s ✉s t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ r❡❧❛t✐♦♥s✿ ✭r❡❧❛t✐♦♥ ✷✮✱ ✭r❡❧❛t✐♦♥ ✸✮✱ ✇❤❡r❡ ✳ ♥❡①t st❡♣✿ ♦❜t❛✐♥ ❛ ❢♦✉rt❤ r❡❧❛t✐♦♥

✶✷ ✴ ✹✶

slide-21
SLIDE 21

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ❉❡r✐✈❛t✐♦♥ ♦❢ ❛ ✶❉ ✜rst✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡

▼♦❞✐✜❝❛t✐♦♥ ♦❢ t❤❡ ❍▲▲ ❛♣♣r♦①✐♠❛t❡ ❘✐❡♠❛♥♥ s♦❧✈❡r

❲❡ ❤❛✈❡ ✹ ✉♥❦♥♦✇♥s t♦ ❞❡t❡r♠✐♥❡✿ W ∗

L =

h∗

L

q∗

L

  • ❛♥❞ W ∗

R =

h∗

R

q∗

R

  • ✳

q ✐s ❛ 0✲❘✐❡♠❛♥♥ ✐♥✈❛r✐❛♥t ✇❡ t❛❦❡ q∗

L = q∗ R = q∗ ✭r❡❧❛t✐♦♥ ✶✮

❚❤❡ ❍❛rt❡♥✲▲❛① ❝♦♥s✐st❡♥❝② ❣✐✈❡s ✉s t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ r❡❧❛t✐♦♥s✿ λRh∗

R − λLh∗ L = (λR − λL)hHLL ✭r❡❧❛t✐♦♥ ✷✮✱

✭r❡❧❛t✐♦♥ ✸✮✱ ✇❤❡r❡ ✳ ♥❡①t st❡♣✿ ♦❜t❛✐♥ ❛ ❢♦✉rt❤ r❡❧❛t✐♦♥

✶✷ ✴ ✹✶

slide-22
SLIDE 22

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ❉❡r✐✈❛t✐♦♥ ♦❢ ❛ ✶❉ ✜rst✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡

▼♦❞✐✜❝❛t✐♦♥ ♦❢ t❤❡ ❍▲▲ ❛♣♣r♦①✐♠❛t❡ ❘✐❡♠❛♥♥ s♦❧✈❡r

❲❡ ❤❛✈❡ ✹ ✉♥❦♥♦✇♥s t♦ ❞❡t❡r♠✐♥❡✿ W ∗

L =

h∗

L

q∗

L

  • ❛♥❞ W ∗

R =

h∗

R

q∗

R

  • ✳

q ✐s ❛ 0✲❘✐❡♠❛♥♥ ✐♥✈❛r✐❛♥t ✇❡ t❛❦❡ q∗

L = q∗ R = q∗ ✭r❡❧❛t✐♦♥ ✶✮

❚❤❡ ❍❛rt❡♥✲▲❛① ❝♦♥s✐st❡♥❝② ❣✐✈❡s ✉s t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ r❡❧❛t✐♦♥s✿ λRh∗

R − λLh∗ L = (λR − λL)hHLL ✭r❡❧❛t✐♦♥ ✷✮✱

q∗ = qHLL + S∆x λR − λL ✭r❡❧❛t✐♦♥ ✸✮✱ ✇❤❡r❡ S ≃ 1 ∆x 1 ∆t ∆x/2

−∆x/2

∆t S(WR(x, t)) dt dx✳ ♥❡①t st❡♣✿ ♦❜t❛✐♥ ❛ ❢♦✉rt❤ r❡❧❛t✐♦♥

✶✷ ✴ ✹✶

slide-23
SLIDE 23

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ❉❡r✐✈❛t✐♦♥ ♦❢ ❛ ✶❉ ✜rst✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡

▼♦❞✐✜❝❛t✐♦♥ ♦❢ t❤❡ ❍▲▲ ❛♣♣r♦①✐♠❛t❡ ❘✐❡♠❛♥♥ s♦❧✈❡r

❲❡ ❤❛✈❡ ✹ ✉♥❦♥♦✇♥s t♦ ❞❡t❡r♠✐♥❡✿ W ∗

L =

h∗

L

q∗

L

  • ❛♥❞ W ∗

R =

h∗

R

q∗

R

  • ✳

q ✐s ❛ 0✲❘✐❡♠❛♥♥ ✐♥✈❛r✐❛♥t ✇❡ t❛❦❡ q∗

L = q∗ R = q∗ ✭r❡❧❛t✐♦♥ ✶✮

❚❤❡ ❍❛rt❡♥✲▲❛① ❝♦♥s✐st❡♥❝② ❣✐✈❡s ✉s t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ r❡❧❛t✐♦♥s✿ λRh∗

R − λLh∗ L = (λR − λL)hHLL ✭r❡❧❛t✐♦♥ ✷✮✱

q∗ = qHLL + S∆x λR − λL ✭r❡❧❛t✐♦♥ ✸✮✱ ✇❤❡r❡ S ≃ 1 ∆x 1 ∆t ∆x/2

−∆x/2

∆t S(WR(x, t)) dt dx✳ ♥❡①t st❡♣✿ ♦❜t❛✐♥ ❛ ❢♦✉rt❤ r❡❧❛t✐♦♥

✶✷ ✴ ✹✶

slide-24
SLIDE 24

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ❉❡r✐✈❛t✐♦♥ ♦❢ ❛ ✶❉ ✜rst✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡

❖❜t❛✐♥✐♥❣ ❛♥ ❛❞❞✐t✐♦♥❛❧ r❡❧❛t✐♦♥

❆ss✉♠❡ t❤❛t WL ❛♥❞ WR ❞❡✜♥❡ ❛ st❡❛❞② st❛t❡✱ ✐✳❡✳ t❤❛t t❤❡② s❛t✐s❢② t❤❡ ❢♦❧❧♦✇✐♥❣ ❞✐s❝r❡t❡ ✈❡rs✐♦♥ ♦❢ t❤❡ st❡❛❞② r❡❧❛t✐♦♥ ∂xF(W) = S(W) ✭✇❤❡r❡ [X] = XR − XL✮✿ 1 ∆x

  • q2

1 h

  • + g

2

  • h2

= S. ❋♦r t❤❡ st❡❛❞② st❛t❡ t♦ ❜❡ ♣r❡s❡r✈❡❞✱ ✐t ✐s s✉✣❝✐❡♥t t♦ ❤❛✈❡ h∗

L = hL✱ h∗ R = hR

❛♥❞ q∗ = q0✳

WL WR λL λR WL WR

❆ss✉♠✐♥❣ ❛ st❡❛❞② st❛t❡✱ ✇❡ s❤♦✇ t❤❛t q∗ = q0✱ ❛s ❢♦❧❧♦✇s✿ q∗ = qHLL + S∆x λR − λL = q0 − 1 λR − λL

  • q2

1 h

  • + g

2

  • h2

− S∆x

  • = q0.

✶✸ ✴ ✹✶

slide-25
SLIDE 25

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ❉❡r✐✈❛t✐♦♥ ♦❢ ❛ ✶❉ ✜rst✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡

❖❜t❛✐♥✐♥❣ ❛♥ ❛❞❞✐t✐♦♥❛❧ r❡❧❛t✐♦♥

■♥ ♦r❞❡r t♦ ❞❡t❡r♠✐♥❡ ❛♥ ❛❞❞✐t✐♦♥❛❧ r❡❧❛t✐♦♥✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ ❞✐s❝r❡t❡ st❡❛❞② r❡❧❛t✐♦♥✱ s❛t✐s✜❡❞ ✇❤❡♥ WL ❛♥❞ WR ❞❡✜♥❡ ❛ st❡❛❞② st❛t❡✿ q2 1 hR − 1 hL

  • + g

2

  • (hR)2 − (hL)2

= S∆x. ❚♦ ❡♥s✉r❡ t❤❛t h∗

L = hL ❛♥❞ h∗ R = hR✱ ✇❡ ✐♠♣♦s❡ t❤❛t h∗ L ❛♥❞ h∗ R

s❛t✐s❢② t❤❡ ❛❜♦✈❡ r❡❧❛t✐♦♥✱ ❛s ❢♦❧❧♦✇s✿ q2 1 h∗

R

− 1 h∗

L

  • + g

2

  • (h∗

R)2 − (h∗ L)2

= S∆x.

✶✹ ✴ ✹✶

slide-26
SLIDE 26

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ❉❡r✐✈❛t✐♦♥ ♦❢ ❛ ✶❉ ✜rst✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡

❉❡t❡r♠✐♥❛t✐♦♥ ♦❢ h∗

L ❛♥❞ h∗ R

❚❤❡ ✐♥t❡r♠❡❞✐❛t❡ ✇❛t❡r ❤❡✐❣❤ts s❛t✐s❢② t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥✿ −q2 h∗

R − h∗ L

h∗

Lh∗ R

  • + g

2(h∗

L + h∗ R)(h∗ R − h∗ L) = S∆x.

❘❡❝❛❧❧ t❤❛t q∗ ✐s ❦♥♦✇♥ ❛♥❞ ✐s ❡q✉❛❧ t♦ q0 ❢♦r ❛ st❡❛❞② st❛t❡✳ ■♥st❡❛❞ ♦❢ t❤❡ ❛❜♦✈❡ r❡❧❛t✐♦♥✱ ✇❡ ❝❤♦♦s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❧✐♥❡❛r✐③❛t✐♦♥✿ −(q∗)2 hLhR (h∗

R − h∗ L) + g

2(hL + hR)(h∗

R − h∗ L) = S∆x,

✇❤✐❝❤ ❝❛♥ ❜❡ r❡✇r✐tt❡♥ ❛s ❢♦❧❧♦✇s✿ −(q∗)2 hLhR + g 2(hL + hR)

  • α

(h∗

R − h∗ L) = S∆x.

✶✺ ✴ ✹✶

slide-27
SLIDE 27

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ❉❡r✐✈❛t✐♦♥ ♦❢ ❛ ✶❉ ✜rst✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡

❉❡t❡r♠✐♥❛t✐♦♥ ♦❢ h∗

L ❛♥❞ h∗ R

❲✐t❤ t❤❡ ❝♦♥s✐st❡♥❝② r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ h∗

L ❛♥❞ h∗ R✱ t❤❡ ✐♥t❡r♠❡❞✐❛t❡

✇❛t❡r ❤❡✐❣❤ts s❛t✐s❢② t❤❡ ❢♦❧❧♦✇✐♥❣ ❧✐♥❡❛r s②st❡♠✿

  • α(h∗

R − h∗ L) = S∆x,

λRh∗

R − λLh∗ L = (λR − λL)hHLL.

❯s✐♥❣ ❜♦t❤ r❡❧❛t✐♦♥s ❧✐♥❦✐♥❣ h∗

L ❛♥❞ h∗ R✱ ✇❡ ♦❜t❛✐♥

         h∗

L = hHLL −

λRS∆x α(λR − λL), h∗

R = hHLL −

λLS∆x α(λR − λL), ✇❤❡r❡ α = −(q∗)2 hLhR + g 2(hL + hR)

  • ✇✐t❤ q∗ = qHLL +

S∆x λR − λL ✳

✶✻ ✴ ✹✶

slide-28
SLIDE 28

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ❉❡r✐✈❛t✐♦♥ ♦❢ ❛ ✶❉ ✜rst✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡

❈♦rr❡❝t✐♦♥ t♦ ❡♥s✉r❡ ♥♦♥✲♥❡❣❛t✐✈❡ h∗

L ❛♥❞ h∗ R

❍♦✇❡✈❡r✱ t❤❡s❡ ❡①♣r❡ss✐♦♥s ♦❢ h∗

L ❛♥❞ h∗ R ❞♦ ♥♦t ❣✉❛r❛♥t❡❡ t❤❛t t❤❡

✐♥t❡r♠❡❞✐❛t❡ ❤❡✐❣❤ts ❛r❡ ♥♦♥✲♥❡❣❛t✐✈❡✿ ✐♥st❡❛❞✱ ✇❡ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝✉t♦✛ ✭s❡❡ ❆✉❞✉ss❡✱ ❈❤❛❧♦♥s✱ ❯♥❣ ✭✷✵✶✺✮✮✿          h∗

L = min

  • hHLL −

λRS∆x α(λR − λL)

  • +

,

  • 1 − λR

λL

  • hHLL
  • ,

h∗

R = min

  • hHLL −

λLS∆x α(λR − λL)

  • +

,

  • 1 − λL

λR

  • hHLL
  • .

◆♦t❡ t❤❛t t❤✐s ❝✉t♦✛ ❞♦❡s ♥♦t ✐♥t❡r❢❡r❡ ✇✐t❤✿ t❤❡ ❝♦♥s✐st❡♥❝② ❝♦♥❞✐t✐♦♥ λRh∗

R − λLh∗ L = (λR − λL)hHLL❀

t❤❡ ✇❡❧❧✲❜❛❧❛♥❝❡ ♣r♦♣❡rt②✱ s✐♥❝❡ ✐t ✐s ♥♦t ❛❝t✐✈❛t❡❞ ✇❤❡♥ WL ❛♥❞ WR ❞❡✜♥❡ ❛ st❡❛❞② st❛t❡✳

✶✼ ✴ ✹✶

slide-29
SLIDE 29

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ❉❡r✐✈❛t✐♦♥ ♦❢ ❛ ✶❉ ✜rst✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡

❙✉♠♠❛r②

❚❤❡ t✇♦✲st❛t❡ ❛♣♣r♦①✐♠❛t❡ ❘✐❡♠❛♥♥ s♦❧✈❡r ✇✐t❤ ✐♥t❡r♠❡❞✐❛t❡ st❛t❡s W ∗

L =

h∗

L

q∗

  • ❛♥❞ W ∗

R =

h∗

R

q∗

  • ❣✐✈❡♥ ❜②

                   q∗ = qHLL + S∆x λR − λL , h∗

L = min

  • hHLL −

λRS∆x α(λR − λL)

  • +

,

  • 1 − λR

λL

  • hHLL
  • ,

h∗

R = min

  • hHLL −

λLS∆x α(λR − λL)

  • +

,

  • 1 − λL

λR

  • hHLL
  • ,

✐s ❝♦♥s✐st❡♥t✱ ♥♦♥✲♥❡❣❛t✐✈✐t②✲♣r❡s❡r✈✐♥❣✱ ❡♥tr♦♣② ♣r❡s❡r✈✐♥❣ ❛♥❞ ✇❡❧❧✲❜❛❧❛♥❝❡❞✳ ♥❡①t st❡♣✿ ❞❡t❡r♠✐♥❛t✐♦♥ ♦❢ S ❛❝❝♦r❞✐♥❣ t♦ t❤❡ s♦✉r❝❡ t❡r♠ ❞❡✜♥✐t✐♦♥ ✭t♦♣♦❣r❛♣❤② ♦r ❢r✐❝t✐♦♥✮✳

✶✽ ✴ ✹✶

slide-30
SLIDE 30

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ❉❡r✐✈❛t✐♦♥ ♦❢ ❛ ✶❉ ✜rst✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡

❚❤❡ t♦♣♦❣r❛♣❤② s♦✉r❝❡ t❡r♠

❲❡ ♥♦✇ ❝♦♥s✐❞❡r S(W) = St(W) = −gh∂xZ✿ t❤❡ s♠♦♦t❤ st❡❛❞② st❛t❡s ❛r❡ ❣♦✈❡r♥❡❞ ❜② ∂x q2 h

  • + g

2∂x

  • h2

= −gh∂xZ, q2 2 ∂x 1 h2

  • + g∂x(h + Z) = 0,

         − − − − − − − →

❞✐s❝r❡t✐③❛t✐♦♥

       q2 1 h

  • + g

2

  • h2

= St∆x, q2 2 1 h2

  • + g[h + Z] = 0.

❲❡ ❝❛♥ ❡①❤✐❜✐t ❛♥ ❡①♣r❡ss✐♦♥ ♦❢ q2

0 ❛♥❞ t❤✉s ♦❜t❛✐♥

St = −g 2hLhR hL + hR [Z] ∆x + g 2∆x [h]3 hL + hR . ❍♦✇❡✈❡r✱ ✇❤❡♥ ZL = ZR✱ ✇❡ ❤❛✈❡ St = O(∆x)✱ ✐✳❡✳ ❛ ❧♦ss ♦❢ ❝♦♥s✐st❡♥❝② ✇✐t❤ St ✭s❡❡ ❢♦r ✐♥st❛♥❝❡ ❇❡rt❤♦♥✱ ❈❤❛❧♦♥s ✭✷✵✶✻✮✮✳

✶✾ ✴ ✹✶

slide-31
SLIDE 31

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ❉❡r✐✈❛t✐♦♥ ♦❢ ❛ ✶❉ ✜rst✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡

❚❤❡ t♦♣♦❣r❛♣❤② s♦✉r❝❡ t❡r♠

■♥st❡❛❞✱ ✇❡ s❡t✱ ❢♦r s♦♠❡ ❝♦♥st❛♥t C > 0✱          St = −g 2hLhR hL + hR [Z] ∆x + g 2∆x [h]3

c

hL + hR , [h]c =

  • hR − hL

✐❢ |hR − hL| ≤ C∆x, sgn(hR − hL) C∆x ♦t❤❡r✇✐s❡. ❚❤❡♦r❡♠✿ ❲❡❧❧✲❜❛❧❛♥❝❡ ❢♦r t❤❡ t♦♣♦❣r❛♣❤② s♦✉r❝❡ t❡r♠ ■❢ WL ❛♥❞ WR ❞❡✜♥❡ ❛ s♠♦♦t❤ st❡❛❞② st❛t❡✱ ✐✳❡✳ ✐❢ t❤❡② s❛t✐s❢② q2 2 1 h2

  • + g[h + Z] = 0,

t❤❡♥ ✇❡ ❤❛✈❡ W ∗

L = WL ❛♥❞ W ∗ R = WR ❛♥❞ t❤❡ ❛♣♣r♦①✐♠❛t❡

❘✐❡♠❛♥♥ s♦❧✈❡r ✐s ✇❡❧❧✲❜❛❧❛♥❝❡❞✳ ❇② ❝♦♥str✉❝t✐♦♥✱ t❤❡ ●♦❞✉♥♦✈✲t②♣❡ s❝❤❡♠❡ ✉s✐♥❣ t❤✐s ❛♣♣r♦①✐♠❛t❡ ❘✐❡♠❛♥♥ s♦❧✈❡r ✐s ❝♦♥s✐st❡♥t✱ ✇❡❧❧✲❜❛❧❛♥❝❡❞✱ ♥♦♥✲♥❡❣❛t✐✈✐t②✲♣r❡s❡r✈✐♥❣ ❛♥❞ ❡♥tr♦♣② ♣r❡s❡r✈✐♥❣✳

✷✵ ✴ ✹✶

slide-32
SLIDE 32

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ❉❡r✐✈❛t✐♦♥ ♦❢ ❛ ✶❉ ✜rst✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡

❚❤❡ ❢r✐❝t✐♦♥ s♦✉r❝❡ t❡r♠

❲❡ ❝♦♥s✐❞❡r✱ ✐♥ t❤✐s ❝❛s❡✱ S(W) = Sf(W) = −kq|q|h−η✱ ✇❤❡r❡ ✇❡ ❤❛✈❡ s❡t η = 7

  • 3✳

❚❤❡ ❛✈❡r❛❣❡ ♦❢ Sf ✇❡ ❝❤♦♦s❡ ✐s Sf = −k¯ q|¯ q|h−η✱ ✇✐t❤ ¯ q t❤❡ ❤❛r♠♦♥✐❝ ♠❡❛♥ ♦❢ qL ❛♥❞ qR ✭♥♦t❡ t❤❛t ¯ q = q0 ❛t t❤❡ ❡q✉✐❧✐❜r✐✉♠✮❀ h−η ❛ ✇❡❧❧✲❝❤♦s❡♥ ❞✐s❝r❡t✐③❛t✐♦♥ ♦❢ h−η✱ ❞❡♣❡♥❞✐♥❣ ♦♥ hL ❛♥❞ hR✱ ❛♥❞ ❡♥s✉r✐♥❣ t❤❡ ✇❡❧❧✲❜❛❧❛♥❝❡ ♣r♦♣❡rt②✳ ❲❡ ❞❡t❡r♠✐♥❡ h−η ✉s✐♥❣ t❤❡ s❛♠❡ t❡❝❤♥✐q✉❡ ✭✇✐t❤ µ0 = sgn(q0)✮✿

∂x q2 h

  • + g

2∂x

  • h2

= −kq0|q0|h−η, q2 ∂xhη−1 η − 1 − g ∂xhη+2 η + 2 = kq0|q0|,        − − − − − − − →

❞✐s❝r❡t✐③❛t✐♦♥

       q2 1 h

  • + g

2

  • h2

= −kµ0q2

0h−η∆x,

q2

  • hη−1

η − 1 − g

  • hη+2

η + 2 = kµ0q2

0∆x.

✷✶ ✴ ✹✶

slide-33
SLIDE 33

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ❉❡r✐✈❛t✐♦♥ ♦❢ ❛ ✶❉ ✜rst✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡

❚❤❡ ❢r✐❝t✐♦♥ s♦✉r❝❡ t❡r♠

❚❤❡ ❡①♣r❡ss✐♦♥ ❢♦r q2

0 ✇❡ ♦❜t❛✐♥❡❞ ✐s ♥♦✇ ✉s❡❞ t♦ ❣❡t✿

h−η = [h2] 2 η + 2 [hη+2] − µ0 k∆x 1 h

  • + [h2]

2 [hη−1] η − 1 η + 2 [hη+2]

  • ,

✇❤✐❝❤ ❣✐✈❡s Sf = −k¯ q|¯ q|h−η ✭h−η ✐s ❝♦♥s✐st❡♥t ✇✐t❤ h−η ✐❢ ❛ ❝✉t♦✛ ✐s ❛♣♣❧✐❡❞ t♦ t❤❡ s❡❝♦♥❞ t❡r♠ ♦❢ h−η✮✳ ❚❤❡♦r❡♠✿ ❲❡❧❧✲❜❛❧❛♥❝❡ ❢♦r t❤❡ ❢r✐❝t✐♦♥ s♦✉r❝❡ t❡r♠ ■❢ WL ❛♥❞ WR ❞❡✜♥❡ ❛ s♠♦♦t❤ st❡❛❞② st❛t❡✱ ✐✳❡✳ ✈❡r✐❢② q2

  • hη−1

η − 1 + g

  • hη+2

η + 2 = −kq0|q0|∆x, t❤❡♥ ✇❡ ❤❛✈❡ W ∗

L = WL ❛♥❞ W ∗ R = WR ❛♥❞ t❤❡ ❛♣♣r♦①✐♠❛t❡

❘✐❡♠❛♥♥ s♦❧✈❡r ✐s ✇❡❧❧✲❜❛❧❛♥❝❡❞✳ ❇② ❝♦♥str✉❝t✐♦♥✱ t❤❡ ●♦❞✉♥♦✈✲t②♣❡ s❝❤❡♠❡ ✉s✐♥❣ t❤✐s ❛♣♣r♦①✐♠❛t❡ ❘✐❡♠❛♥♥ s♦❧✈❡r ✐s ❝♦♥s✐st❡♥t✱ ✇❡❧❧✲❜❛❧❛♥❝❡❞✱ ♥♦♥✲♥❡❣❛t✐✈✐t②✲♣r❡s❡r✈✐♥❣ ❛♥❞ ❡♥tr♦♣② ♣r❡s❡r✈✐♥❣✳

✷✷ ✴ ✹✶

slide-34
SLIDE 34

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ❉❡r✐✈❛t✐♦♥ ♦❢ ❛ ✶❉ ✜rst✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡

❋r✐❝t✐♦♥ ❛♥❞ t♦♣♦❣r❛♣❤② s♦✉r❝❡ t❡r♠s

❲✐t❤ ❜♦t❤ s♦✉r❝❡ t❡r♠s✱ t❤❡ s❝❤❡♠❡ ♣r❡s❡r✈❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❞✐s❝r❡t✐③❛t✐♦♥ ♦❢ t❤❡ st❡❛❞② r❡❧❛t✐♦♥ ∂xF(W) = S(W)✿ q2 1 h

  • + g

2

  • h2

= St∆x + Sf∆x. ❚❤❡ ✐♥t❡r♠❡❞✐❛t❡ st❛t❡s ❛r❡ t❤❡r❡❢♦r❡ ❣✐✈❡♥ ❜②✿                    q∗ = qHLL + (St + Sf)∆x λR − λL ; h∗

L = min

  • hHLL − λR(St + Sf)∆x

α(λR − λL)

  • +

,

  • 1 − λR

λL

  • hHLL
  • ;

h∗

R = min

  • hHLL − λL(St + Sf)∆x

α(λR − λL)

  • +

,

  • 1 − λL

λR

  • hHLL
  • .

✷✸ ✴ ✹✶

slide-35
SLIDE 35

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ❉❡r✐✈❛t✐♦♥ ♦❢ ❛ ✶❉ ✜rst✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡

❚❤❡ ❢✉❧❧ ●♦❞✉♥♦✈✲t②♣❡ s❝❤❡♠❡

x t tn+1 tn xi xi− 1

2

xi+ 1

2

W n

i

W R,∗

i− 1

2

W L,∗

i+ 1

2

λR

i− 1

2

λL

i+ 1

2

  • W ∆(x, tn+1)

❲❡ r❡❝❛❧❧ W n+1

i

= 1 ∆x xi+ 1

2

xi− 1

2

W ∆(x, tn+1)dx✿ t❤❡♥ W n+1

i

= W n

i − ∆t

∆x

  • λL

i+ 1

2

  • W L,∗

i+ 1

2 − W n

i

  • − λR

i− 1

2

  • W R,∗

i− 1

2 − W n

i

  • ,

✇❤✐❝❤ ❝❛♥ ❜❡ r❡✇r✐tt❡♥✱ ❛❢t❡r str❛✐❣❤t❢♦r✇❛r❞ ❝♦♠♣✉t❛t✐♦♥s✱

W n+1

i

= W n

i − ∆t

∆x

  • F n

i+ 1

2 − F n

i− 1

2

  • + ∆t

    (St)n

i− 1

2+(St)n

i+ 1

2

2  +   (Sf)n

i− 1

2+(Sf)n

i+ 1

2

2    .

✷✹ ✴ ✹✶

slide-36
SLIDE 36

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ❉❡r✐✈❛t✐♦♥ ♦❢ ❛ ✶❉ ✜rst✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡

❙✉♠♠❛r②

❲❡ ❤❛✈❡ ♣r❡s❡♥t❡❞ ❛ s❝❤❡♠❡ t❤❛t✿ ✐s ❝♦♥s✐st❡♥t ✇✐t❤ t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ ❢r✐❝t✐♦♥ ❛♥❞ t♦♣♦❣r❛♣❤②❀ ✐s ✇❡❧❧✲❜❛❧❛♥❝❡❞ ❢♦r ❢r✐❝t✐♦♥ ❛♥❞ t♦♣♦❣r❛♣❤② st❡❛❞② st❛t❡s❀ ♣r❡s❡r✈❡s t❤❡ ♥♦♥✲♥❡❣❛t✐✈✐t② ♦❢ t❤❡ ✇❛t❡r ❤❡✐❣❤t❀ ❡♥s✉r❡s ❛ ❞✐s❝r❡t❡ ❡♥tr♦♣② ✐♥❡q✉❛❧✐t②❀ ✐s ♥♦t ❛❜❧❡ t♦ ❝♦rr❡❝t❧② ❛♣♣r♦①✐♠❛t❡ ✇❡t✴❞r② ✐♥t❡r❢❛❝❡s ❞✉❡ t♦ t❤❡ st✐✛♥❡ss ♦❢ t❤❡ ❢r✐❝t✐♦♥ kq|q|h−7

  • 3✿ t❤❡ ❢r✐❝t✐♦♥ t❡r♠ s❤♦✉❧❞ ❜❡

tr❡❛t❡❞ ✐♠♣❧✐❝✐t❧②✳ ♥❡①t st❡♣✿ ✐♥tr♦❞✉❝t✐♦♥ ♦❢ t❤✐s s❡♠✐✲✐♠♣❧✐❝✐t s❝❤❡♠❡

✷✺ ✴ ✹✶

slide-37
SLIDE 37

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ❉❡r✐✈❛t✐♦♥ ♦❢ ❛ ✶❉ ✜rst✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡

❙❡♠✐✲✐♠♣❧✐❝✐t ✜♥✐t❡ ✈♦❧✉♠❡ s❝❤❡♠❡

❲❡ ✉s❡ ❛ s♣❧✐tt✐♥❣ ♠❡t❤♦❞ ✇✐t❤ ❛♥ ❡①♣❧✐❝✐t tr❡❛t♠❡♥t ♦❢ t❤❡ ✢✉① ❛♥❞ t❤❡ t♦♣♦❣r❛♣❤② ❛♥❞ ❛♥ ✐♠♣❧✐❝✐t tr❡❛t♠❡♥t ♦❢ t❤❡ ❢r✐❝t✐♦♥✳

✶ ❡①♣❧✐❝✐t❧② s♦❧✈❡ ∂tW + ∂xF(W) = St(W) ❛s ❢♦❧❧♦✇s✿

W

n+ 1

2

i

= W n

i − ∆t

∆x

  • Fn

i+ 1

2 − Fn

i− 1

2

  • + ∆t
  • 1

2

  • (St)n

i− 1

2 + (St)n

i+ 1

2

  • ✷ ✐♠♣❧✐❝✐t❧② s♦❧✈❡ ∂tW = Sf(W) ❛s ❢♦❧❧♦✇s✿

         hn+1

i

= h

n+ 1

2

i

■❱P✿ ∂tq = −kq|q|(hn+1

i

)−η q(xi, tn) = q

n+ 1

2

i

qn+1

i

✷✻ ✴ ✹✶

slide-38
SLIDE 38

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ❉❡r✐✈❛t✐♦♥ ♦❢ ❛ ✶❉ ✜rst✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡

❙❡♠✐✲✐♠♣❧✐❝✐t ✜♥✐t❡ ✈♦❧✉♠❡ s❝❤❡♠❡

❙♦❧✈✐♥❣ t❤❡ ■❱P ②✐❡❧❞s✿ qn+1

i

= (hn+1

i

)ηq

n+ 1

2

i

(hn+1

i

)η + k ∆t

  • q

n+ 1

2

i

  • .

❲❡ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ (hn+1

i

)η✱ ✇❤✐❝❤ ♣r♦✈✐❞❡s ✉s ✇✐t❤ ❛♥ ❡①♣r❡ss✐♦♥ ♦❢ qn+1

i

t❤❛t ✐s ❡q✉❛❧ t♦ q0 ❛t t❤❡ ❡q✉✐❧✐❜r✐✉♠✿ (hη)n+1

i

= 2µ

n+ 1

2

i

µn

i

  • h−ηn+1

i− 1

2

+

  • h−ηn+1

i+ 1

2

+ k ∆t µ

n+ 1

2

i

qn

i .

s❡♠✐✲✐♠♣❧✐❝✐t tr❡❛t♠❡♥t ♦❢ t❤❡ ❢r✐❝t✐♦♥ s♦✉r❝❡ t❡r♠ s❝❤❡♠❡ ❛❜❧❡ t♦ ♠♦❞❡❧ ✇❡t✴❞r② tr❛♥s✐t✐♦♥s s❝❤❡♠❡ st✐❧❧ ✇❡❧❧✲❜❛❧❛♥❝❡❞ ❛♥❞ ♥♦♥✲♥❡❣❛t✐✈✐t②✲♣r❡s❡r✈✐♥❣

✷✼ ✴ ✹✶

slide-39
SLIDE 39

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ❚✇♦✲❞✐♠❡♥s✐♦♥❛❧ ❛♥❞ ❤✐❣❤✲♦r❞❡r ❡①t❡♥s✐♦♥s

✶ ■♥tr♦❞✉❝t✐♦♥ t♦ ●♦❞✉♥♦✈✲t②♣❡ s❝❤❡♠❡s ✷ ❉❡r✐✈❛t✐♦♥ ♦❢ ❛ ✶❉ ✜rst✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ✸ ❚✇♦✲❞✐♠❡♥s✐♦♥❛❧ ❛♥❞ ❤✐❣❤✲♦r❞❡r ❡①t❡♥s✐♦♥s ✹ ✷❉ ❛♥❞ ❤✐❣❤✲♦r❞❡r ♥✉♠❡r✐❝❛❧ s✐♠✉❧❛t✐♦♥s ✺ ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ♣❡rs♣❡❝t✐✈❡s

slide-40
SLIDE 40

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ❚✇♦✲❞✐♠❡♥s✐♦♥❛❧ ❛♥❞ ❤✐❣❤✲♦r❞❡r ❡①t❡♥s✐♦♥s

❚✇♦✲❞✐♠❡♥s✐♦♥❛❧ ❡①t❡♥s✐♦♥

✷❉ s❤❛❧❧♦✇✲✇❛t❡r ♠♦❞❡❧✿ ∂tW + ∇ · F (W) = St(W) + Sf(W)        ∂th + ∇ · q = 0 ∂tq + ∇ · q ⊗ q h + 1 2gh2I2

  • = −gh∇Z − kqq

hη t♦ t❤❡ r✐❣❤t✿ s✐♠✉❧❛t✐♦♥ ♦❢ t❤❡ ✷✵✶✶ ❏❛♣❛♥ ts✉♥❛♠✐

✷✽ ✴ ✹✶

slide-41
SLIDE 41

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ❚✇♦✲❞✐♠❡♥s✐♦♥❛❧ ❛♥❞ ❤✐❣❤✲♦r❞❡r ❡①t❡♥s✐♦♥s

❚✇♦✲❞✐♠❡♥s✐♦♥❛❧ ❡①t❡♥s✐♦♥

s♣❛❝❡ ❞✐s❝r❡t✐③❛t✐♦♥✿ ❈❛rt❡s✐❛♥ ♠❡s❤

xi ci eij cj nij

❲✐t❤ Fn

ij = F(W n i , W n j ; nij) ❛♥❞ νi t❤❡ ♥❡✐❣❤❜♦rs ♦❢ ci✱ t❤❡ s❝❤❡♠❡ r❡❛❞s✿

W

n+ 1

2

i

= W n

i − ∆t

  • j∈νi

|eij| |ci| Fn

ij + ∆t

2

  • j∈νi

(St)n

ij.

W n+1

i

✐s ♦❜t❛✐♥❡❞ ❢r♦♠ W

n+ 1

2

i

✇✐t❤ ❛ s♣❧✐tt✐♥❣ str❛t❡❣②✿

  • ∂th = 0

∂tq = −k qqh−η          hn+1

i

= h

n+ 1

2

i

qn+1

i

= (hη)n+1

i

q

n+ 1

2

i

(hη)n+1

i

+ k ∆t

  • q

n+ 1

2

i

  • ✷✾ ✴ ✹✶
slide-42
SLIDE 42

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ❚✇♦✲❞✐♠❡♥s✐♦♥❛❧ ❛♥❞ ❤✐❣❤✲♦r❞❡r ❡①t❡♥s✐♦♥s

❚✇♦✲❞✐♠❡♥s✐♦♥❛❧ ❡①t❡♥s✐♦♥

❚❤❡ ✷❉ s❝❤❡♠❡ ✐s✿ ♥♦♥✲♥❡❣❛t✐✈✐t②✲♣r❡s❡r✈✐♥❣ ❢♦r t❤❡ ✇❛t❡r ❤❡✐❣❤t✿ ∀i ∈ Z, hn

i ≥ 0 =

⇒ ∀i ∈ Z, hn+1

i

≥ 0❀ ❛❜❧❡ t♦ ❞❡❛❧ ✇✐t❤ ✇❡t✴❞r② tr❛♥s✐t✐♦♥s t❤❛♥❦s t♦ t❤❡ s❡♠✐✲✐♠♣❧✐❝✐t❛t✐♦♥ ✇✐t❤ t❤❡ s♣❧✐tt✐♥❣ ♠❡t❤♦❞❀ ✇❡❧❧✲❜❛❧❛♥❝❡❞ ❜② ❞✐r❡❝t✐♦♥ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ ❢r✐❝t✐♦♥ ❛♥❞✴♦r t♦♣♦❣r❛♣❤②✱ ✐✳❡✳✿

✐t ♣r❡s❡r✈❡s ❛❧❧ st❡❛❞② st❛t❡s ❛t r❡st✱ ✐t ♣r❡s❡r✈❡s ❢r✐❝t✐♦♥ ❛♥❞✴♦r t♦♣♦❣r❛♣❤② st❡❛❞② st❛t❡s ✐♥ t❤❡ x✲❞✐r❡❝t✐♦♥ ❛♥❞ t❤❡ y✲❞✐r❡❝t✐♦♥✱ ✐t ❞♦❡s ♥♦t ♣r❡s❡r✈❡ t❤❡ ❢✉❧❧② ✷❉ st❡❛❞② st❛t❡s✳

♥❡①t st❡♣✿ ❤✐❣❤✲♦r❞❡r ❡①t❡♥s✐♦♥ ♦❢ t❤✐s ✷❉ s❝❤❡♠❡

✸✵ ✴ ✹✶

slide-43
SLIDE 43

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ❚✇♦✲❞✐♠❡♥s✐♦♥❛❧ ❛♥❞ ❤✐❣❤✲♦r❞❡r ❡①t❡♥s✐♦♥s

❍✐❣❤✲♦r❞❡r ❡①t❡♥s✐♦♥✿ t❤❡ ❜❛s✐❝s✱ ✐♥ ✶❉

xi−1 xi+1 xi W n

i+1

W n

i−1

W n

i

x xi+ 1

2

xi− 1

2

W n

i ∈ P0✿ ❝♦♥st❛♥t ✭♦r❞❡r ✶ s❝❤❡♠❡✮

✸✶ ✴ ✹✶

slide-44
SLIDE 44

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ❚✇♦✲❞✐♠❡♥s✐♦♥❛❧ ❛♥❞ ❤✐❣❤✲♦r❞❡r ❡①t❡♥s✐♦♥s

❍✐❣❤✲♦r❞❡r ❡①t❡♥s✐♦♥✿ t❤❡ ❜❛s✐❝s✱ ✐♥ ✶❉

xi−1 xi+1 xi W n

i+1

W n

i−1

W n

i

  • W n

i (x)

W n

i,−

W n

i,+

x xi+ 1

2

xi− 1

2

  • W n

i ∈ P1✿ ❧✐♥❡❛r ✭♦r❞❡r ✷ s❝❤❡♠❡✮

✸✶ ✴ ✹✶

slide-45
SLIDE 45

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ❚✇♦✲❞✐♠❡♥s✐♦♥❛❧ ❛♥❞ ❤✐❣❤✲♦r❞❡r ❡①t❡♥s✐♦♥s

❍✐❣❤✲♦r❞❡r ❡①t❡♥s✐♦♥✿ t❤❡ ❜❛s✐❝s✱ ✐♥ ✶❉

xi−1 xi+1 xi W n

i+1

W n

i−1

W n

i

  • W n

i (x)

W n

i,−

W n

i,+

x xi+ 1

2

xi− 1

2

  • W n

i ∈ Pd✿ ♣♦❧②♥♦♠✐❛❧ ✭♦r❞❡r d + 1 s❝❤❡♠❡✮

✸✶ ✴ ✹✶

slide-46
SLIDE 46

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ❚✇♦✲❞✐♠❡♥s✐♦♥❛❧ ❛♥❞ ❤✐❣❤✲♦r❞❡r ❡①t❡♥s✐♦♥s

❍✐❣❤✲♦r❞❡r ❡①t❡♥s✐♦♥✿ t❤❡ ♣♦❧②♥♦♠✐❛❧ r❡❝♦♥str✉❝t✐♦♥

♣♦❧②♥♦♠✐❛❧ r❡❝♦♥str✉❝t✐♦♥ ✭s❡❡ ❉✐♦t✱ ❈❧❛✐♥✱ ▲♦✉❜èr❡ ✭✷✵✶✷✮✮✿

  • W n

i (x) = W n i + d

  • |k|=1

αk

i

  • (x − xi)k − Mk

i

  • ❲❡ ❤❛✈❡ Mk

i =

1 |ci|

  • ci

(x − xi)kdx s✉❝❤ t❤❛t t❤❡ ❝♦♥s❡r✈❛t✐♦♥ ♣r♦♣❡rt② ✐s ✈❡r✐✜❡❞✿ 1 |ci|

  • ci
  • W n

i (x)dx = W n i ✳ xi W n

i+1

W n

i−1

W n

i

  • W n

i (x)

x xi+ 1

2

xi− 1

2

ci ∈ S2

i

/ ∈ S2

i

❚❤❡ ♣♦❧②♥♦♠✐❛❧ ❝♦❡✣❝✐❡♥ts αk

i ❛r❡ ❝❤♦s❡♥ t♦ ♠✐♥✐♠✐③❡ t❤❡ ❧❡❛st sq✉❛r❡s

❡rr♦r ❜❡t✇❡❡♥ t❤❡ r❡❝♦♥str✉❝t✐♦♥ ❛♥❞ W n

j ✱ ❢♦r ❛❧❧ j ✐♥ t❤❡ st❡♥❝✐❧ Sd i ✳

✸✷ ✴ ✹✶

slide-47
SLIDE 47

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ❚✇♦✲❞✐♠❡♥s✐♦♥❛❧ ❛♥❞ ❤✐❣❤✲♦r❞❡r ❡①t❡♥s✐♦♥s

❍✐❣❤✲♦r❞❡r ❡①t❡♥s✐♦♥✿ t❤❡ s❝❤❡♠❡

❍✐❣❤✲♦r❞❡r s♣❛❝❡ ❛❝❝✉r❛❝② W n+1

i

= W n

i −∆t

  • j∈νi

|eij| |ci|

R

  • r=0

ξrFn

ij,r+∆t Q

  • q=0

ηq

  • (St)n

i,q + (Sf)n i,q

  • Fn

ij,r = F(

W n

i (σr),

W n

j (σr); nij)

(St)n

i,q = St(

W n

i (xq))

❛♥❞ (Sf)n

i,q = Sf(

W n

i (xq))

❲❡ ❤❛✈❡ s❡t✿ (ξr, σr)r✱ ❛ q✉❛❞r❛t✉r❡ r✉❧❡ ♦♥ t❤❡ ❡❞❣❡ eij❀ (ηq, xq)q✱ ❛ q✉❛❞r❛t✉r❡ r✉❧❡ ♦♥ t❤❡ ❝❡❧❧ ci✳ ❚❤❡ ❤✐❣❤✲♦r❞❡r t✐♠❡ ❛❝❝✉r❛❝② ✐s ❛❝❤✐❡✈❡❞ ❜② t❤❡ ✉s❡ ♦❢ ❙❙P❘❑ ♠❡t❤♦❞s ✭s❡❡ ●♦tt❧✐❡❜✱ ❙❤✉ ✭✶✾✾✽✮✮✳

✸✸ ✴ ✹✶

slide-48
SLIDE 48

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ❚✇♦✲❞✐♠❡♥s✐♦♥❛❧ ❛♥❞ ❤✐❣❤✲♦r❞❡r ❡①t❡♥s✐♦♥s

❲❡❧❧✲❜❛❧❛♥❝❡ r❡❝♦✈❡r② ✭✶❉✮✿ ❛ ❝♦♥✈❡① ❝♦♠❜✐♥❛t✐♦♥

r❡❝♦♥str✉❝t✐♦♥ ♣r♦❝❡❞✉r❡ t❤❡ s❝❤❡♠❡ ♥♦ ❧♦♥❣❡r ♣r❡s❡r✈❡s st❡❛❞② st❛t❡s ❲❡❧❧✲❜❛❧❛♥❝❡ r❡❝♦✈❡r② ❲❡ s✉❣❣❡st ❛ ❝♦♥✈❡① ❝♦♠❜✐♥❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❤✐❣❤✲♦r❞❡r s❝❤❡♠❡ WHO ❛♥❞ t❤❡ ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ WWB✿ W n+1

i

= θn

i (WHO)n+1 i

+ (1 − θn

i )(WWB)n+1 i

, ✇✐t❤ θn

i t❤❡ ♣❛r❛♠❡t❡r ♦❢ t❤❡ ❝♦♥✈❡① ❝♦♠❜✐♥❛t✐♦♥✱ s✉❝❤ t❤❛t✿

✐❢ θn

i = 0✱ t❤❡♥ t❤❡ ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ✐s ✉s❡❞❀

✐❢ θn

i = 1✱ t❤❡♥ t❤❡ ❤✐❣❤✲♦r❞❡r s❝❤❡♠❡ ✐s ✉s❡❞✳

♥❡①t st❡♣✿ ❞❡r✐✈❡ ❛ s✉✐t❛❜❧❡ ❡①♣r❡ss✐♦♥ ❢♦r θn

i

✸✹ ✴ ✹✶

slide-49
SLIDE 49

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ❚✇♦✲❞✐♠❡♥s✐♦♥❛❧ ❛♥❞ ❤✐❣❤✲♦r❞❡r ❡①t❡♥s✐♦♥s

❲❡❧❧✲❜❛❧❛♥❝❡ r❡❝♦✈❡r② ✭✶❉✮✿ ❛ st❡❛❞② st❛t❡ ❞❡t❡❝t♦r

❙t❡❛❞② st❛t❡ ❞❡t❡❝t♦r st❡❛❞② st❛t❡ s♦❧✉t✐♦♥✿    qL = qR = q0, E := q2 hR − q2 hL + g 2

  • h2

R − h2 L

  • − (St + Sf)∆x = 0

st❡❛❞② st❛t❡ ❞❡t❡❝t♦r✿ ϕn

i =

qn

i − qn i−1

[E]n

i− 1

2

 

  • 2

+

qn

i+1 − qn i

[E]n

i+ 1

2

 

  • 2

ϕn

i = 0 ✐❢ t❤❡r❡ ✐s ❛ st❡❛❞② st❛t❡

❜❡t✇❡❡♥ W n

i−1✱ W n i ❛♥❞ W n i+1

✐♥ t❤✐s ❝❛s❡✱ ✇❡ t❛❦❡ θn

i = 0

♦t❤❡r✇✐s❡✱ ✇❡ t❛❦❡ 0 < θn

i ≤ 1

1 m∆x M∆x θn

i

ϕn

i

WB HO

✸✺ ✴ ✹✶

slide-50
SLIDE 50

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ❚✇♦✲❞✐♠❡♥s✐♦♥❛❧ ❛♥❞ ❤✐❣❤✲♦r❞❡r ❡①t❡♥s✐♦♥s

▼❖❖❉ ♠❡t❤♦❞

❍✐❣❤✲♦r❞❡r s❝❤❡♠❡s ✐♥❞✉❝❡ ♦s❝✐❧❧❛t✐♦♥s✿ ✇❡ ✉s❡ t❤❡ ▼❖❖❉ ♠❡t❤♦❞ t♦ ❣❡t r✐❞ ♦❢ t❤❡ ♦s❝✐❧❧❛t✐♦♥s ❛♥❞ t♦ r❡st♦r❡ t❤❡ ♥♦♥✲♥❡❣❛t✐✈✐t② ♣r❡s❡r✈❛t✐♦♥ ✭s❡❡ ❈❧❛✐♥✱ ❉✐♦t✱ ▲♦✉❜èr❡ ✭✷✵✶✶✮✮✳ ▼❖❖❉ ❧♦♦♣

✶ ❝♦♠♣✉t❡ ❛ ❝❛♥❞✐❞❛t❡ s♦❧✉t✐♦♥ W c ✇✐t❤ t❤❡ ❤✐❣❤✲♦r❞❡r s❝❤❡♠❡ ✷ ❞❡t❡r♠✐♥❡ ✇❤❡t❤❡r W c ✐s ❛❞♠✐ss✐❜❧❡✱ ✐✳❡✳

✐❢ hc ✐s ♥♦♥✲♥❡❣❛t✐✈❡ ✭P❆❉ ❝r✐t❡r✐♦♥✮ ✐❢ W c ❞♦❡s ♥♦t ♣r❡s❡♥t s♣✉r✐♦✉s ♦s❝✐❧❧❛t✐♦♥s ✭❉▼P ❛♥❞ ✉✷ ❝r✐t❡r✐❛✮

✸ ✇❤❡r❡ ♥❡❝❡ss❛r②✱ ❞❡❝r❡❛s❡ t❤❡ ❞❡❣r❡❡ ♦❢ t❤❡ r❡❝♦♥str✉❝t✐♦♥ ✹ ❝♦♠♣✉t❡ ❛ ♥❡✇ ❝❛♥❞✐❞❛t❡ s♦❧✉t✐♦♥

✸✻ ✴ ✹✶

slide-51
SLIDE 51

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ✷❉ ❛♥❞ ❤✐❣❤✲♦r❞❡r ♥✉♠❡r✐❝❛❧ s✐♠✉❧❛t✐♦♥s

✶ ■♥tr♦❞✉❝t✐♦♥ t♦ ●♦❞✉♥♦✈✲t②♣❡ s❝❤❡♠❡s ✷ ❉❡r✐✈❛t✐♦♥ ♦❢ ❛ ✶❉ ✜rst✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ✸ ❚✇♦✲❞✐♠❡♥s✐♦♥❛❧ ❛♥❞ ❤✐❣❤✲♦r❞❡r ❡①t❡♥s✐♦♥s ✹ ✷❉ ❛♥❞ ❤✐❣❤✲♦r❞❡r ♥✉♠❡r✐❝❛❧ s✐♠✉❧❛t✐♦♥s ✺ ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ♣❡rs♣❡❝t✐✈❡s

slide-52
SLIDE 52

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ✷❉ ❛♥❞ ❤✐❣❤✲♦r❞❡r ♥✉♠❡r✐❝❛❧ s✐♠✉❧❛t✐♦♥s

Ps❡✉❞♦✲✶❉ ❞♦✉❜❧❡ ❞r② ❞❛♠✲❜r❡❛❦ ♦♥ ❛ s✐♥✉s♦✐❞❛❧ ❜♦tt♦♠

❚❤❡ P❲❇

5

s❝❤❡♠❡ ✐s ✉s❡❞ ✐♥ t❤❡ ✇❤♦❧❡ ❞♦♠❛✐♥✿ ♥❡❛r t❤❡ ❜♦✉♥❞❛r✐❡s✱ st❡❛❞② st❛t❡ ❛t r❡st ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡❀ ❛✇❛② ❢r♦♠ t❤❡ ❜♦✉♥❞❛r✐❡s✱ ❢❛r ❢r♦♠ st❡❛❞② st❛t❡ ❤✐❣❤✲♦r❞❡r s❝❤❡♠❡❀ ❝❡♥t❡r✱ ❞r② ❛r❡❛ ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡✳

✸✼ ✴ ✹✶

slide-53
SLIDE 53

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ✷❉ ❛♥❞ ❤✐❣❤✲♦r❞❡r ♥✉♠❡r✐❝❛❧ s✐♠✉❧❛t✐♦♥s

❖r❞❡r ♦❢ ❛❝❝✉r❛❝② ❛ss❡ss♠❡♥t

❚♦ ❛ss❡ss t❤❡ ♦r❞❡r ♦❢ ❛❝❝✉r❛❝②✱ ✇❡ t❛❦❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡①❛❝t st❡❛❞② s♦❧✉t✐♦♥ ♦❢ t❤❡ ✷❉ s❤❛❧❧♦✇✲✇❛t❡r s②st❡♠✱ ✇❤❡r❡ r = t(x, y)✿ h = 1 ; q = r r ; Z = 2kr − 1 2gr2 . ❲✐t❤ k = 10✱ t❤✐s s♦❧✉t✐♦♥ ✐s ❞❡♣✐❝t❡❞ ❜❡❧♦✇ ♦♥ t❤❡ s♣❛❝❡ ❞♦♠❛✐♥ [−0.3, 0.3] × [0.4, 1]✳

✸✽ ✴ ✹✶

slide-54
SLIDE 54

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ✷❉ ❛♥❞ ❤✐❣❤✲♦r❞❡r ♥✉♠❡r✐❝❛❧ s✐♠✉❧❛t✐♦♥s

❖r❞❡r ♦❢ ❛❝❝✉r❛❝② ❛ss❡ss♠❡♥t

L2 ❡rr♦rs ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ♥✉♠❜❡r ♦❢ ❝❡❧❧s t♦♣ ❣r❛♣❤s✿ ✷❉ st❡❛❞② s♦❧✉t✐♦♥ ✇✐t❤ t♦♣♦❣r❛♣❤② ❜♦tt♦♠ ❣r❛♣❤s✿ ✷❉ st❡❛❞② s♦❧✉t✐♦♥ ✇✐t❤ ❢r✐❝t✐♦♥ ❛♥❞ t♦♣♦❣r❛♣❤②

1e3 1e4 1e-4 1e-6 1e-8 1e-10 4 6 1 h, PWB

3

h, PWB

5

1e3 1e4 1e-4 1e-6 1e-8 1e-10 4 6 1 q, PWB

3

q, PWB

5

1e3 1e4 1e-6 1e-8 1e-10 1e-12 4 6 1 h, PWB

3

h, PWB

5

1e3 1e4 1e-6 1e-8 1e-10 1e-12 4 6 1 q, PWB

3

q, PWB

5

✸✽ ✴ ✹✶

slide-55
SLIDE 55

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ✷❉ ❛♥❞ ❤✐❣❤✲♦r❞❡r ♥✉♠❡r✐❝❛❧ s✐♠✉❧❛t✐♦♥s

✷✵✶✶ ❚✠ ♦❤♦❦✉ ts✉♥❛♠✐

❚s✉♥❛♠✐ s✐♠✉❧❛t✐♦♥ ♦♥ ❛ ❈❛rt❡s✐❛♥ ♠❡s❤✿ ✶✸ ♠✐❧❧✐♦♥ ❝❡❧❧s✱ ❋♦rtr❛♥ ❝♦❞❡ ♣❛r❛❧❧❡❧✐③❡❞ ✇✐t❤ ❖♣❡♥▼P✱ r✉♥ ♦♥ ✹✽ ❝♦r❡s✳

✸✾ ✴ ✹✶

slide-56
SLIDE 56

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ✷❉ ❛♥❞ ❤✐❣❤✲♦r❞❡r ♥✉♠❡r✐❝❛❧ s✐♠✉❧❛t✐♦♥s

✷✵✶✶ ❚✠ ♦❤♦❦✉ ts✉♥❛♠✐

500 1,000 1,500 2,000 2,500 −8 −6 −4 −2 Russia (Vladivostok) Sea of Japan Japan (Hokkaid¯

  • island)

Kuril trench Pacific Ocean 1D slice of the topography (unit: kilometers).

✸✾ ✴ ✹✶

slide-57
SLIDE 57

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ✷❉ ❛♥❞ ❤✐❣❤✲♦r❞❡r ♥✉♠❡r✐❝❛❧ s✐♠✉❧❛t✐♦♥s

✷✵✶✶ ❚✠ ♦❤♦❦✉ ts✉♥❛♠✐

✸✾ ✴ ✹✶

slide-58
SLIDE 58

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ✷❉ ❛♥❞ ❤✐❣❤✲♦r❞❡r ♥✉♠❡r✐❝❛❧ s✐♠✉❧❛t✐♦♥s

✷✵✶✶ ❚✠ ♦❤♦❦✉ ts✉♥❛♠✐

♣❤②s✐❝❛❧ t✐♠❡ ♦❢ t❤❡ s✐♠✉❧❛t✐♦♥✿ 1 ❤♦✉r ✜rst✲♦r❞❡r s❝❤❡♠❡ ❈P❯ t✐♠❡✿ ∼ 1.1 ❤♦✉r s❡❝♦♥❞✲♦r❞❡r s❝❤❡♠❡ ❈P❯ t✐♠❡✿ ∼ 2.7 ❤♦✉rs

✸✾ ✴ ✹✶

slide-59
SLIDE 59

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ✷❉ ❛♥❞ ❤✐❣❤✲♦r❞❡r ♥✉♠❡r✐❝❛❧ s✐♠✉❧❛t✐♦♥s

✷✵✶✶ ❚✠ ♦❤♦❦✉ ts✉♥❛♠✐

❲❛t❡r ❞❡♣t❤ ❛t t❤❡ s❡♥s♦rs✿ ★✶✿ ✺✼✵✵ ♠❀ ★✷✿ ✻✶✵✵ ♠❀ ★✸✿ ✹✹✵✵ ♠✳

  • r❛♣❤s ♦❢ t❤❡ t✐♠❡ ✈❛r✐❛t✐♦♥

♦❢ t❤❡ ✇❛t❡r ❤❡✐❣❤t ✭✐♥ ♠❡t❡rs✮✳ ❞❛t❛ ✐♥ ❜❧❛❝❦✱ ♦r❞❡r ✶ ✐♥ ❜❧✉❡✱ ♦r❞❡r ✷ ✐♥ r❡❞ 1,200 2,400 3,600

−0.2 0.2 0.4 0.6 Sensor #1 1,200 2,400 3,600 0.1 0.2 Sensor #2 1,200 2,400 3,600 0.1 0.2 Sensor #3

✸✾ ✴ ✹✶

slide-60
SLIDE 60

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ♣❡rs♣❡❝t✐✈❡s

✶ ■♥tr♦❞✉❝t✐♦♥ t♦ ●♦❞✉♥♦✈✲t②♣❡ s❝❤❡♠❡s ✷ ❉❡r✐✈❛t✐♦♥ ♦❢ ❛ ✶❉ ✜rst✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ✸ ❚✇♦✲❞✐♠❡♥s✐♦♥❛❧ ❛♥❞ ❤✐❣❤✲♦r❞❡r ❡①t❡♥s✐♦♥s ✹ ✷❉ ❛♥❞ ❤✐❣❤✲♦r❞❡r ♥✉♠❡r✐❝❛❧ s✐♠✉❧❛t✐♦♥s ✺ ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ♣❡rs♣❡❝t✐✈❡s

slide-61
SLIDE 61

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ♣❡rs♣❡❝t✐✈❡s

❈♦♥❝❧✉s✐♦♥

❲❡ ❤❛✈❡ ♣r❡s❡♥t❡❞ ❛ ✇❡❧❧✲❜❛❧❛♥❝❡❞✱ ♥♦♥✲♥❡❣❛t✐✈✐t②✲♣r❡s❡r✈✐♥❣ ❛♥❞ ❡♥✲ tr♦♣② ♣r❡s❡r✈✐♥❣ ♥✉♠❡r✐❝❛❧ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥✱ ❛❜❧❡ t♦ ❜❡ ❛♣♣❧✐❡❞ t♦ ♦t❤❡r s♦✉r❝❡ t❡r♠s ♦r ❝♦♠❜✐♥❛t✐♦♥s ♦❢ s♦✉r❝❡ t❡r♠s✳ ❲❡ ❤❛✈❡ ❛❧s♦ ❞✐s♣❧❛②❡❞ r❡s✉❧ts ❢r♦♠ t❤❡ ✷❉ ❤✐❣❤✲♦r❞❡r ❡①t❡♥s✐♦♥ ♦❢ t❤✐s ♥✉♠❡r✐❝❛❧ ♠❡t❤♦❞✱ ❝♦❞❡❞ ✐♥ ❋♦rtr❛♥ ❛♥❞ ♣❛r❛❧❧❡❧✐③❡❞ ✇✐t❤ ❖♣❡♥▼P✳

❚❤✐s ✇♦r❦ ❤❛s ❜❡❡♥ ♣✉❜❧✐s❤❡❞✿ ❱✳ ▼✳✲❉✳✱ ❈✳ ❇❡rt❤♦♥✱ ❙✳ ❈❧❛✐♥ ❛♥❞ ❋✳ ❋♦✉❝❤❡r✳ ✏❆ ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤②✑✳ ❈♦♠♣✉t✳ ▼❛t❤✳ ❆♣♣❧✳ ✼✷✭✸✮✿✺✻✽✕✺✾✸✱ ✷✵✶✻✳ ❱✳ ▼✳✲❉✳✱ ❈✳ ❇❡rt❤♦♥✱ ❙✳ ❈❧❛✐♥ ❛♥❞ ❋✳ ❋♦✉❝❤❡r✳ ✏❆ ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ♦r ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥✑✳ ❏✳ ❈♦♠♣✉t✳ P❤②s✳ ✸✸✺✿✶✶✺✕✶✺✹✱ ✷✵✶✼✳ ❈✳ ❇❡rt❤♦♥✱ ❘✳ ▲♦✉❜èr❡✱ ❛♥❞ ❱✳ ▼✳✲❉✳ ✏❆ s❡❝♦♥❞✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤②✑✳ ❆❝❝❡♣t❡❞ ✐♥ ❙♣r✐♥❣❡r Pr♦❝✳ ▼❛t❤✳ ❙t❛t✳✱ ✷✵✶✼✳ ❈✳ ❇❡rt❤♦♥ ❛♥❞ ❱✳ ▼✳✲❉✳ ✏❆ s✐♠♣❧❡ ❢✉❧❧② ✇❡❧❧✲❜❛❧❛♥❝❡❞ ❛♥❞ ❡♥tr♦♣② ♣r❡s❡r✈✐♥❣ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s✑✳ ❙✉❜♠✐tt❡❞✳

✹✵ ✴ ✹✶

slide-62
SLIDE 62

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ♣❡rs♣❡❝t✐✈❡s

P❡rs♣❡❝t✐✈❡s

❲♦r❦ ✐♥ ♣r♦❣r❡ss ❤✐❣❤✲♦r❞❡r s✐♠✉❧❛t✐♦♥ ♦❢ t❤❡ ✷✵✶✶ ❚✠ ♦❤♦❦✉ ts✉♥❛♠✐ ❛♣♣❧✐❝❛t✐♦♥ t♦ ♦t❤❡r s♦✉r❝❡ t❡r♠s✿ ❈♦r✐♦❧✐s ❢♦r❝❡ s♦✉r❝❡ t❡r♠ ❜r❡❛❞t❤ ✈❛r✐❛t✐♦♥ s♦✉r❝❡ t❡r♠ ▲♦♥❣✲t❡r♠ ♣❡rs♣❡❝t✐✈❡s ❡♥s✉r❡ t❤❡ ❡♥tr♦♣② ♣r❡s❡r✈❛t✐♦♥ ❢♦r t❤❡ ❤✐❣❤✲♦r❞❡r s❝❤❡♠❡ ✭✉s❡ ♦❢ ❛♥ ❡✲▼❖❖❉ ♠❡t❤♦❞✮ s✐♠✉❧❛t✐♦♥ ♦❢ r♦❣✉❡ ✇❛✈❡s

✹✶ ✴ ✹✶

slide-63
SLIDE 63

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ❚❤❛♥❦s✦

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦

slide-64
SLIDE 64

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ❆♣♣❡♥❞✐❝❡s

❚❤❡ ❞✐s❝r❡t❡ ❡♥tr♦♣② ✐♥❡q✉❛❧✐t②

❚❤❡ ❢♦❧❧♦✇✐♥❣ ♥♦♥✲❝♦♥s❡r✈❛t✐✈❡ ❡♥tr♦♣② ✐♥❡q✉❛❧✐t② ✐s s❛t✐s✜❡❞ ❜② t❤❡ s❤❛❧❧♦✇✲✇❛t❡r s②st❡♠✿ ∂tη(W) + ∂xG(W) ≤ q hS(W); η(W) = q2 2h + gh2 2 ; G(W) = q h q2 2h + gh2

  • .

❆t t❤❡ ❞✐s❝r❡t❡ ❧❡✈❡❧✱ ✇❡ s❤♦✇ t❤❛t✿ λR(η∗

R − ηR) − λL(η∗ L − ηL)+(GR − GL) ≤ qHLL

hHLL S∆x+O(∆x2). ♠❛✐♥ ✐♥❣r❡❞✐❡♥ts✿ h∗

L = hHLL − S∆x

λR α(λR − λL) ✭❛♥❞ s✐♠✐❧❛r ❡①♣r❡ss✐♦♥s ❢♦r h∗

R ❛♥❞ q∗✮

(λR − λL)ηHLL ≤ λRηR − λLηL − (GR − GL) ❢r♦♠ ❍❛rt❡♥✱ ▲❛①✱ ✈❛♥ ▲❡❡r ✭✶✾✽✸✮

slide-65
SLIDE 65

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ❆♣♣❡♥❞✐❝❡s

❱❡r✐✜❝❛t✐♦♥ ♦❢ t❤❡ ✇❡❧❧✲❜❛❧❛♥❝❡✿ t♦♣♦❣r❛♣❤②

❚❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ✐s ❛t r❡st❀ ✇❛t❡r ✐s ✐♥❥❡❝t❡❞ t❤r♦✉❣❤ t❤❡ ❧❡❢t ❜♦✉♥❞❛r②✳

slide-66
SLIDE 66

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ❆♣♣❡♥❞✐❝❡s

❱❡r✐✜❝❛t✐♦♥ ♦❢ t❤❡ ✇❡❧❧✲❜❛❧❛♥❝❡✿ t♦♣♦❣r❛♣❤②

❚❤❡ ♥♦♥✲✇❡❧❧✲❜❛❧❛♥❝❡❞ ❍▲▲ s❝❤❡♠❡ ❝♦♥✈❡r❣❡s t♦✇❛r❞s ❛ ♥✉♠❡r✐❝❛❧ st❡❛❞② st❛t❡ ✇❤✐❝❤ ❞♦❡s ♥♦t ❝♦rr❡s♣♦♥❞ t♦ t❤❡ ♣❤②s✐❝❛❧ ♦♥❡✳

slide-67
SLIDE 67

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ❆♣♣❡♥❞✐❝❡s

❱❡r✐✜❝❛t✐♦♥ ♦❢ t❤❡ ✇❡❧❧✲❜❛❧❛♥❝❡✿ t♦♣♦❣r❛♣❤②

❚❤❡ ♥♦♥✲✇❡❧❧✲❜❛❧❛♥❝❡❞ ❍▲▲ s❝❤❡♠❡ ②✐❡❧❞s ❛ ♥✉♠❡r✐❝❛❧ st❡❛❞② st❛t❡ ✇❤✐❝❤ ❞♦❡s ♥♦t ❝♦rr❡s♣♦♥❞ t♦ t❤❡ ♣❤②s✐❝❛❧ ♦♥❡✳ ❚❤❡ ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❡①❛❝t❧② ②✐❡❧❞s t❤❡ ♣❤②s✐❝❛❧ st❡❛❞② st❛t❡✳

slide-68
SLIDE 68

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ❆♣♣❡♥❞✐❝❡s

❱❡r✐✜❝❛t✐♦♥ ♦❢ t❤❡ ✇❡❧❧✲❜❛❧❛♥❝❡✿ t♦♣♦❣r❛♣❤②

❚❤❡ ♥♦♥✲✇❡❧❧✲❜❛❧❛♥❝❡❞ ❍▲▲ s❝❤❡♠❡ ②✐❡❧❞s ❛ ♥✉♠❡r✐❝❛❧ st❡❛❞② st❛t❡ ✇❤✐❝❤ ❞♦❡s ♥♦t ❝♦rr❡s♣♦♥❞ t♦ t❤❡ ♣❤②s✐❝❛❧ ♦♥❡✳ ❚❤❡ ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❡①❛❝t❧② ②✐❡❧❞s t❤❡ ♣❤②s✐❝❛❧ st❡❛❞② st❛t❡✳

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SLIDE 69

❆ ❤✐❣❤✲♦r❞❡r ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❝❤❡♠❡ ❢♦r t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ t♦♣♦❣r❛♣❤② ❛♥❞ ▼❛♥♥✐♥❣ ❢r✐❝t✐♦♥ ❆♣♣❡♥❞✐❝❡s

❱❡r✐✜❝❛t✐♦♥ ♦❢ t❤❡ ✇❡❧❧✲❜❛❧❛♥❝❡✿ t♦♣♦❣r❛♣❤②

tr❛♥s❝r✐t✐❝❛❧ ✢♦✇ t❡st ❝❛s❡ ✭s❡❡ ●♦✉t❛❧✱ ▼❛✉r❡❧ ✭✶✾✾✼✮✮ ❧❡❢t ♣❛♥❡❧✿ ✐♥✐t✐❛❧ ❢r❡❡ s✉r❢❛❝❡ ❛t r❡st❀ ✇❛t❡r ✐s ✐♥❥❡❝t❡❞ ❢r♦♠ t❤❡ ❧❡❢t ❜♦✉♥❞❛r② r✐❣❤t ♣❛♥❡❧✿ ❢r❡❡ s✉r❢❛❝❡ ❢♦r t❤❡ st❡❛❞② st❛t❡ s♦❧✉t✐♦♥✱ ❛❢t❡r ❛ tr❛♥s✐❡♥t st❛t❡ Φ = u2 2 + g(h + Z) L1 L2 L∞ ❡rr♦rs ♦♥ q ✶✳✹✼❡✲✶✹ ✶✳✺✽❡✲✶✹ ✷✳✵✹❡✲✶✹ ❡rr♦rs ♦♥ Φ ✶✳✻✼❡✲✶✹ ✷✳✶✸❡✲✶✹ ✹✳✷✻❡✲✶✹