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❆ ❤✐❣❤✲♦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

❆ ❤✐❣❤✲♦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✉❣❛❧✱ ✷✵✶✵✮ ✶ ✴ ✹✶

❆ ❤✐❣❤✲♦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   ∂ t h + ∂ x ( hu ) = 0  � � hu 2 + 1 = − gh∂ x Z − kq | q |  2 gh 2 ∂ t ( hu ) + ∂ x ✭✇✐t❤ q = hu )  h 7 � 3 � h � ❲❡ ❝❛♥ r❡✇r✐t❡ t❤❡ ❡q✉❛t✐♦♥s ❛s ∂ t W + ∂ x F ( W ) = S ( W ) ✱ ✇✐t❤ W = ✳ q Z ( x ) ✐s t❤❡ ❦♥♦✇♥ water surface t♦♣♦❣r❛♣❤② k ✐s t❤❡ ▼❛♥♥✐♥❣ h ( x, t ) u ( x, t ) ❝♦❡✣❝✐❡♥t channel bottom Z ( x ) g ✐s t❤❡ ❣r❛✈✐t❛t✐♦♥❛❧ ❝♦♥st❛♥t x ✇❡ ❧❛❜❡❧ t❤❡ ✇❛t❡r ❞✐s❝❤❛r❣❡ q := hu ✷ ✴ ✹✶

❆ ❤✐❣❤✲♦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✐♦♥ ✐✛ ∂ t W = 0 ✱ ✐✳❡✳ ∂ x F ( W ) = S ( W ) ✳ ❚❛❦✐♥❣ ∂ t W = 0 ✐♥ t❤❡ s❤❛❧❧♦✇✲✇❛t❡r ❡q✉❛t✐♦♥s ❧❡❛❞s t♦   ∂ x q = 0  � q 2 � h + 1 = − gh∂ x Z − kq | q |  2 gh 2 ∂ x .  h 7 � 3 ❚❤❡ st❡❛❞② st❛t❡ s♦❧✉t✐♦♥s ❛r❡ t❤❡r❡❢♦r❡ ❣✐✈❡♥ ❜②   q = cst = q 0  � q 2 � h + 1 = − gh∂ x Z − kq 0 | q 0 |  0 2 gh 2 ∂ x .  h 7 � 3 ✸ ✴ ✹✶

❆ ❤✐❣❤✲♦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❝❤❡♠❡✦ ✹ ✴ ✹✶

❆ ❤✐❣❤✲♦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❝❤❡♠❡✦ ✹ ✴ ✹✶

❆ ❤✐❣❤✲♦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❤✮✳ ✺ ✴ ✹✶

❆ ❤✐❣❤✲♦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

❆ ❤✐❣❤✲♦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❡♠ ∂ t W + ∂ x F ( W ) = S ( W ) ✱ ✇✐t❤ s✉✐t❛❜❧❡ ✐♥✐t✐❛❧ ❛♥❞ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✳ ❲❡ ♣❛rt✐t✐♦♥ t❤❡ s♣❛❝❡ ❞♦♠❛✐♥ ✐♥ ❝❡❧❧s ✱ ♦❢ ✈♦❧✉♠❡ ∆ x ❛♥❞ ♦❢ ❡✈❡♥❧② s♣❛❝❡❞ ❝❡♥t❡rs x i ✱ ❛♥❞ ✇❡ ❞❡✜♥❡✿ x i − 1 2 ❛♥❞ x i + 1 2 ✱ t❤❡ ❜♦✉♥❞❛r✐❡s ♦❢ t❤❡ ❝❡❧❧ i ❀ W n i ✱ ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ W ( x, t ) ✱ ❝♦♥st❛♥t ✐♥ t❤❡ ❝❡❧❧ i ❛♥❞ � ∆ x/ 2 1 W ( x, t n ) dx ✳ ❛t t✐♠❡ t n ✱ ✇❤✐❝❤ ✐s ❞❡✜♥❡❞ ❛s W n i = ∆ x ∆ x/ 2 W n W ( x, t ) i x x x i − 1 x i x i +1 x i − 1 x i + 1 2 2 ✻ ✴ ✹✶