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❆❞❛♣t✐✈❡ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t ✭❆P❘✮ ❛♣♣❧✐❡❞ t♦ ❙P❍ ♠❡t❤♦❞ ✿ ❙t❛❜✐❧✐t②✱ ❛❝❝✉r❛❝② ❛♥❞ ❈P❯ ❛♥❛❧②s✐s ▲❛✉r❡♥t ❈❤✐r♦♥ ▲❍❊❊❆ ▲❛❜ ✲ ❊❝♦❧❡ ❈❡♥tr❛❧❡ ◆❛♥t❡s ✶ ❡r ❥✉✐❧❧❡t ✷✵✶✺ ✶ ❡r ❥✉✐❧❧❡t ✷✵✶✺ ▲❛✉r❡♥t ❈❤✐r♦♥ ❆❞❛♣t✐✈❡ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t ✭❆P❘✮ ❛♣♣❧✐❡❞ t♦ ❙P❍ ♠❡t❤♦❞ ✿ ❙t❛❜✐❧✐t②✱ ❛❝❝✉r❛❝② ❛♥❞ ❈P❯ ❛♥❛❧②s✐s ✶ ✴ ✷✺

P❧❛♥ ✶ ❙P❍ ❜❛❝❦❣r♦✉♥❞ ✷ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t t❤❡♦r② ✸ P❛rt✐❝❧❡ r❡✜♥❡♠❡♥t ✐♠♣r♦✈❡♠❡♥ts ❆r❜✐tr❛r② ▲❛❣r❛♥❣✐❛♥ ❊✉❧❡r✐❛♥ ✭❆▲❊✮ ♣r♦❝❡ss ❚❤❡ ❆P❘ ❝♦♥❝❡♣t P❛r❛❧❧❡❧✐③❛t✐♦♥ ✹ ❙♦♠❡ r❡s✉❧ts ❑❧❡❡❢s♠❛♥ ❞❛♠❜r❡❛❦ ✶ ❡r ❥✉✐❧❧❡t ✷✵✶✺ ▲❛✉r❡♥t ❈❤✐r♦♥ ❆❞❛♣t✐✈❡ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t ✭❆P❘✮ ❛♣♣❧✐❡❞ t♦ ❙P❍ ♠❡t❤♦❞ ✿ ❙t❛❜✐❧✐t②✱ ❛❝❝✉r❛❝② ❛♥❞ ❈P❯ ❛♥❛❧②s✐s ✷ ✴ ✷✺

P❧❛♥ ✶ ❙P❍ ❜❛❝❦❣r♦✉♥❞ ✷ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t t❤❡♦r② ✸ P❛rt✐❝❧❡ r❡✜♥❡♠❡♥t ✐♠♣r♦✈❡♠❡♥ts ❆r❜✐tr❛r② ▲❛❣r❛♥❣✐❛♥ ❊✉❧❡r✐❛♥ ✭❆▲❊✮ ♣r♦❝❡ss ❚❤❡ ❆P❘ ❝♦♥❝❡♣t P❛r❛❧❧❡❧✐③❛t✐♦♥ ✹ ❙♦♠❡ r❡s✉❧ts ❑❧❡❡❢s♠❛♥ ❞❛♠❜r❡❛❦ ✶ ❡r ❥✉✐❧❧❡t ✷✵✶✺ ▲❛✉r❡♥t ❈❤✐r♦♥ ❆❞❛♣t✐✈❡ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t ✭❆P❘✮ ❛♣♣❧✐❡❞ t♦ ❙P❍ ♠❡t❤♦❞ ✿ ❙t❛❜✐❧✐t②✱ ❛❝❝✉r❛❝② ❛♥❞ ❈P❯ ❛♥❛❧②s✐s ✸ ✴ ✷✺

❙♣❤✲✢♦✇ s❝♦♣❡s ✶ ❡r ❥✉✐❧❧❡t ✷✵✶✺ ▲❛✉r❡♥t ❈❤✐r♦♥ ❆❞❛♣t✐✈❡ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t ✭❆P❘✮ ❛♣♣❧✐❡❞ t♦ ❙P❍ ♠❡t❤♦❞ ✿ ❙t❛❜✐❧✐t②✱ ❛❝❝✉r❛❝② ❛♥❞ ❈P❯ ❛♥❛❧②s✐s ✹ ✴ ✷✺

❍♦✇ t♦ ❝❛❧❝✉❧❛t❡ t❤❡ t❡r♠ ✐♥ ❙P❍ ❄ ❈♦♥✈♦❧✉t✐♦♥ ♣r♦❞✉❝t ♣r♦♣❡rt✐❡s ✐s ❛♣♣r♦❛❝❤❡❞ ❜② ❛ ❦❡r♥❡❧ ❢✉♥❝t✐♦♥ ❙P❍ ❜❛❝❦❣r♦✉♥❞ ❈♦♥s❡r✈❛t✐♦♥ ❧❛✇  ❋✐♥✐t❡ ❞✐✛❡r❡♥❝❡s   ∂ t U + ∂ x f ( U ) = 0 ❋✐♥✐t❡ ✈♦❧✉♠❡s  ∂ x f ( U ) → ❉✐s❝♦♥t✐♥✉♦✉s ●❛❧❡r❦✐♥   ❙P❍  ✶ ❡r ❥✉✐❧❧❡t ✷✵✶✺ ▲❛✉r❡♥t ❈❤✐r♦♥ ❆❞❛♣t✐✈❡ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t ✭❆P❘✮ ❛♣♣❧✐❡❞ t♦ ❙P❍ ♠❡t❤♦❞ ✿ ❙t❛❜✐❧✐t②✱ ❛❝❝✉r❛❝② ❛♥❞ ❈P❯ ❛♥❛❧②s✐s ✺ ✴ ✷✺

❈♦♥✈♦❧✉t✐♦♥ ♣r♦❞✉❝t ♣r♦♣❡rt✐❡s ✐s ❛♣♣r♦❛❝❤❡❞ ❜② ❛ ❦❡r♥❡❧ ❢✉♥❝t✐♦♥ ❙P❍ ❜❛❝❦❣r♦✉♥❞ ❈♦♥s❡r✈❛t✐♦♥ ❧❛✇  ❋✐♥✐t❡ ❞✐✛❡r❡♥❝❡s   ∂ t U + ∂ x f ( U ) = 0 ❋✐♥✐t❡ ✈♦❧✉♠❡s  ∂ x f ( U ) → ❉✐s❝♦♥t✐♥✉♦✉s ●❛❧❡r❦✐♥   ❙P❍  • ❍♦✇ t♦ ❝❛❧❝✉❧❛t❡ t❤❡ t❡r♠ ∂ x f ( U ) ✐♥ ❙P❍ ❄ ✶ ❡r ❥✉✐❧❧❡t ✷✵✶✺ ▲❛✉r❡♥t ❈❤✐r♦♥ ❆❞❛♣t✐✈❡ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t ✭❆P❘✮ ❛♣♣❧✐❡❞ t♦ ❙P❍ ♠❡t❤♦❞ ✿ ❙t❛❜✐❧✐t②✱ ❛❝❝✉r❛❝② ❛♥❞ ❈P❯ ❛♥❛❧②s✐s ✺ ✴ ✷✺

❙P❍ ❜❛❝❦❣r♦✉♥❞ ❈♦♥s❡r✈❛t✐♦♥ ❧❛✇  ❋✐♥✐t❡ ❞✐✛❡r❡♥❝❡s   ∂ t U + ∂ x f ( U ) = 0 ❋✐♥✐t❡ ✈♦❧✉♠❡s  ∂ x f ( U ) → ❉✐s❝♦♥t✐♥✉♦✉s ●❛❧❡r❦✐♥   ❙P❍  • ❍♦✇ t♦ ❝❛❧❝✉❧❛t❡ t❤❡ t❡r♠ ∂ x f ( U ) ✐♥ ❙P❍ ❄ ❈♦♥✈♦❧✉t✐♦♥ ♣r♦❞✉❝t ♣r♦♣❡rt✐❡s � f ∗ δ = R f ( y ) δ ( x − y ) dy = f ∇ f ∗ g = f ∗ ∇ g δ ✐s ❛♣♣r♦❛❝❤❡❞ ❜② ❛ ❦❡r♥❡❧ ❢✉♥❝t✐♦♥ W ∇ f i ≈ ∇ f ∗ W = f ∗ ∇ W = � w j f j ∇ W ij j ✶ ❡r ❥✉✐❧❧❡t ✷✵✶✺ ▲❛✉r❡♥t ❈❤✐r♦♥ ❆❞❛♣t✐✈❡ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t ✭❆P❘✮ ❛♣♣❧✐❡❞ t♦ ❙P❍ ♠❡t❤♦❞ ✿ ❙t❛❜✐❧✐t②✱ ❛❝❝✉r❛❝② ❛♥❞ ❈P❯ ❛♥❛❧②s✐s ✺ ✴ ✷✺

❙P❍ ❜❛❝❦❣r♦✉♥❞ ❇❡♥❡✜ts • ▼❡s❤❧❡ss ✭♣❛rt✐❝❧❡s✮ • ▲❛❣r❛♥❣✐❛♥ ♠❡t❤♦❞ ❲❡❛❝❦♥❡ss❡s • ▲♦✇ s♣❛t✐❛❧ ♦r❞❡r • ◆♦ ❝♦♥✈❡r❣❡♥❝❡ t❤❡♦r❡♠ ✧❈♦♥✈❡r❣❡♥❝❡✧ ❝♦♥❞✐t✐♦♥s h • ∆ x → ∞ ❛♥❞ h → 0 ✶ ❡r ❥✉✐❧❧❡t ✷✵✶✺ ▲❛✉r❡♥t ❈❤✐r♦♥ ❆❞❛♣t✐✈❡ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t ✭❆P❘✮ ❛♣♣❧✐❡❞ t♦ ❙P❍ ♠❡t❤♦❞ ✿ ❙t❛❜✐❧✐t②✱ ❛❝❝✉r❛❝② ❛♥❞ ❈P❯ ❛♥❛❧②s✐s ✻ ✴ ✷✺

P❧❛♥ ✶ ❙P❍ ❜❛❝❦❣r♦✉♥❞ ✷ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t t❤❡♦r② ✸ P❛rt✐❝❧❡ r❡✜♥❡♠❡♥t ✐♠♣r♦✈❡♠❡♥ts ❆r❜✐tr❛r② ▲❛❣r❛♥❣✐❛♥ ❊✉❧❡r✐❛♥ ✭❆▲❊✮ ♣r♦❝❡ss ❚❤❡ ❆P❘ ❝♦♥❝❡♣t P❛r❛❧❧❡❧✐③❛t✐♦♥ ✹ ❙♦♠❡ r❡s✉❧ts ❑❧❡❡❢s♠❛♥ ❞❛♠❜r❡❛❦ ✶ ❡r ❥✉✐❧❧❡t ✷✵✶✺ ▲❛✉r❡♥t ❈❤✐r♦♥ ❆❞❛♣t✐✈❡ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t ✭❆P❘✮ ❛♣♣❧✐❡❞ t♦ ❙P❍ ♠❡t❤♦❞ ✿ ❙t❛❜✐❧✐t②✱ ❛❝❝✉r❛❝② ❛♥❞ ❈P❯ ❛♥❛❧②s✐s ✼ ✴ ✷✺

P❛rt✐❝❧❡ r❡✜♥❡♠❡♥t t❤❡♦r② ❙♦♠❡ ♣❛r❛♠❡t❡rs • ❙♣❛❝✐♥❣ ♣❛r❛♠❡t❡r ǫ ∈ [0 , 1] • ❘❛❞✐✉s r❛t✐♦ α ∈ ]0 , 1] • ❙✇✐t❝❤ ♣❛r❛♠❡t❡r γ ∈ [0 , 1] ǫ ∆ x m • • • • • • ∆ x m • • • • • • • • • • • • ❘❡✜♥❡♠❡♥t • • • • • • • R m • • • R m • • Pr♦❝❡ss • • • • • • αR m • • • • • • • • • • • • • • • • • • ✶ ❡r ❥✉✐❧❧❡t ✷✵✶✺ ▲❛✉r❡♥t ❈❤✐r♦♥ ❆❞❛♣t✐✈❡ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t ✭❆P❘✮ ❛♣♣❧✐❡❞ t♦ ❙P❍ ♠❡t❤♦❞ ✿ ❙t❛❜✐❧✐t②✱ ❛❝❝✉r❛❝② ❛♥❞ ❈P❯ ❛♥❛❧②s✐s ✽ ✴ ✷✺

P❛rt✐❝❧❡ r❡✜♥❡♠❡♥t t❤❡♦r② ❙✇✐t❝❤ ♣❛r❛♠❡t❡r ♣r♦♣❡rt✐❡s • γ = 1 → ♣❛rt✐❝❧❡ ✐s t✉r♥❡❞ ♦♥ ✭❛❝t✐✈❡ ♣❛rt✐❝❧❡✮ • γ = 0 → ♣❛rt✐❝❧❡ ✐s t✉r♥❡❞ ♦✛ ✭♣❛ss✐✈❡ ♣❛rt✐❝❧❡✮ γ γ m γ f 1 ❘❡✜♥❡❞ ❛r❡❛ ▼♦❞✐✜❡❞ ❙P❍ ♦♣❡r❛t♦r • �∇ f � i = � f j ω j ∇ W ij γ j 0 ❇✉✛❡r ③♦♥❡ ❇✉✛❡r ③♦♥❡ j ✶ ❡r ❥✉✐❧❧❡t ✷✵✶✺ ▲❛✉r❡♥t ❈❤✐r♦♥ ❆❞❛♣t✐✈❡ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t ✭❆P❘✮ ❛♣♣❧✐❡❞ t♦ ❙P❍ ♠❡t❤♦❞ ✿ ❙t❛❜✐❧✐t②✱ ❛❝❝✉r❛❝② ❛♥❞ ❈P❯ ❛♥❛❧②s✐s ✾ ✴ ✷✺

P❧❛♥ ✶ ❙P❍ ❜❛❝❦❣r♦✉♥❞ ✷ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t t❤❡♦r② ✸ P❛rt✐❝❧❡ r❡✜♥❡♠❡♥t ✐♠♣r♦✈❡♠❡♥ts ❆r❜✐tr❛r② ▲❛❣r❛♥❣✐❛♥ ❊✉❧❡r✐❛♥ ✭❆▲❊✮ ♣r♦❝❡ss ❚❤❡ ❆P❘ ❝♦♥❝❡♣t P❛r❛❧❧❡❧✐③❛t✐♦♥ ✹ ❙♦♠❡ r❡s✉❧ts ❑❧❡❡❢s♠❛♥ ❞❛♠❜r❡❛❦ ✶ ❡r ❥✉✐❧❧❡t ✷✵✶✺ ▲❛✉r❡♥t ❈❤✐r♦♥ ❆❞❛♣t✐✈❡ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t ✭❆P❘✮ ❛♣♣❧✐❡❞ t♦ ❙P❍ ♠❡t❤♦❞ ✿ ❙t❛❜✐❧✐t②✱ ❛❝❝✉r❛❝② ❛♥❞ ❈P❯ ❛♥❛❧②s✐s ✶✵ ✴ ✷✺