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senior_thesis_slides

 senior_thesis_slides

statictaku

May 20, 2023
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  1. Reformulation of Density-Independent Smoothed Particle
    Hydrodynamics with Riemann Solver: Godunov DISPH
    ᴷ ৽ SPH εΩʔϜ:Godunov DISPH ๏ʹ͍ͭͯ ᴷ
    to be submitted to MNRAS
    1 ౬ઙ୓޺
    ࢦಋڭһ: 1 ৿ਖ਼෉
    ஜ೾େֶ
    ଔݚ಺༰֬ೝ, January 26, 2023
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 1 / 45

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  2. ໨࣍
    1 ֤छεΩʔϜʹ͍ͭͯ
    Standard SPH ๏
    SSPH with Artificial Conductivity ๏
    Density-Independent SPH ๏
    Godunov SPH ๏
    ·ͱΊ
    2 Godunov DISPH ๏ಋग़
    3 ςετܭࢉ
    Riemann ໰୊
    2D ѹྗฏߧ
    ఺ݯരൃ
    2D Kelvin-Helmholtz ෆ҆ఆੑ
    4 ݁࿦
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 2 / 45

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  3. ֤छεΩʔϜʹ͍ͭͯ Standard SPH ๏
    Standard SPH (SSPH) ๏
    SSPH ๏ಋग़ʹ༻͍Δඇ೪ੑஅ೤ѹॖੑྲྀମͷӡಈํఔࣜɺΤωϧΪʔํఔࣜ
    dv(r)
    dt
    = −
    1
    ρ(r)
    ∇P(r) + FAV (αAV ) (1)
    du(r)
    dt
    = −
    P(r)
    ρ(r)
    ∇ · v(r) + GAV (αAV ) (2)
    ཻࢠ i ͕࣋ͭ೚ҙͷ෺ཧྔ
    F(ri) =

    F(r′
    )W(|ri − r′
    |, h)d3r′
    + O(h2)
    =

    j
    mj
    ρj
    FjW(|ri − rj|, h) + O(1)
    (3)
    ρi =

    j
    mjWij(h) (4)
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 3 / 45

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  4. ֤छεΩʔϜʹ͍ͭͯ Standard SPH ๏
    • σϝϦοτ
    ▶ ྲྀମͷ઀৮ෆ࿈ଓ໘Λ͏·͘ѻ͑ͣɺද໘ுྗతͳޮՌ͕ݱΕͯ͠·͏
    ▶ িܸ೾Λଊ͑ΔͨΊʹਓ޻తͳࢄҳ߲͕ӡಈํఔࣜɺΤωϧΪʔํఔࣜʹඞཁɻ
    ਓ޻೪ੑͷڧ͞Λௐઅ͢Δ೚ҙύϥϝʔλ α ΛਓؒͷखͰௐ੔͢Δඞཁ
    ▶ ཭ࢄԽͷࡍʹཻࢠ෼෍ͷඇ౳ํੑ͔Β͘Δۭؒθϩ࣍ͷޡ͕ࠩൃੜ
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 4 / 45

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  5. ֤छεΩʔϜʹ͍ͭͯ Standard SPH ๏
    ઀৮ෆ࿈ଓ໘
    • ີ౓͸Χʔωϧิ׬ʹΑͬͯٻΊΒΕΔ
    ρ =

    j
    mjWij(h) (5)
    • ಺෦ΤωϧΪʔ͸࣌ؒੵ෼ͰٻΊΒΕΔ
    • ઀৮ෆ࿈ଓ໘Ͱɺີ౓͸ͳΊΒ͔ɺ಺෦
    ΤωϧΪʔ͸ٸܹʹมԽ
    P = (γ − 1)ρu (6)
    ▶ P ʹෆ੔߹͕ੜ͡ɺͦͷ݁Ռද໘ுྗ͕
    ൃੜ [Price, 2008]
    ▶ ཻࢠͷମੵཁૉ dV = m
    ρ
    ͸ີ౓ෆ࿈ଓ
    ͷͱ͜ΖͰਫ਼౓௿Լ
    [Saitoh and Makino, 2013]
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 5 / 45

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  6. ֤छεΩʔϜʹ͍ͭͯ Standard SPH ๏
    • ߋʹɺಋग़தʹີ౓ͷඍ෼Λ࢖༻

    ∇P
    ρ
    = −∇
    (
    P
    ρ
    )

    P
    ρ2
    ∇ρ (7)
    ෺ཧతʹີ౓ෆ࿈ଓͱͳΔ઀৮ෆ࿈ଓ໘Ͱ͸ɺີ౓ͷۭؒඍ෼Λܦ༝ͯ͠ಋग़͞Εͨࣜ
    ͸ਖ਼͘͠ͳ͍͸ͣ [Saitoh and Makino, 2013]
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 6 / 45

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  7. ֤छεΩʔϜʹ͍ͭͯ Standard SPH ๏
    Riemann ໰୊
    {
    ρ = 1, P = 1, v = 0 x < 0 ͷͱ͖
    ρ = 0.125, P = 0.1, v = 0 x > 0 ͷͱ͖
    (8)
    ີ౓ ѹྗ ଎౓
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 7 / 45

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  8. ֤छεΩʔϜʹ͍ͭͯ Standard SPH ๏
    Riemann ໰୊
    ڧ͍িܸ೾͕ൃੜ͢Δ໰୊
    {
    ρ = 1, P = 1000, v = 0 x < 0 ͷͱ͖
    ρ = 1, P = 0.1, v = 0 x > 0 ͷͱ͖
    (9)
    ີ౓ ѹྗ ଎౓
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 8 / 45

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  9. ֤छεΩʔϜʹ͍ͭͯ Standard SPH ๏
    SSPH
    িܸ೾؅໰୊ ڧ͍িܸ೾ (ॳظ৚݅Ͱ 104 ͷѹྗࠩ)
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 9 / 45

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  10. ֤छεΩʔϜʹ͍ͭͯ Standard SPH ๏
    2D ѹྗฏߧ
    ॳظ৚݅ͷ··มಈ͠ͳ͍ͷ͕෺ཧతͳղ ѹྗ͸શྖҬͰҰఆɹ
    • Ի଎ Cs = 1.02 ͰܭࢉྖҬΛԣ੾Δ࣌ؒ໿ 1.0 ͷ 8 ഒͰ͋Δ t = 8.0 ·Ͱܭࢉ
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 10 / 45

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  11. ֤छεΩʔϜʹ͍ͭͯ Standard SPH ๏
    • ௿ີ౓ྖҬͷԻ଎ Cs = 1.02, ߴີ౓ྖҬͷԻ଎ Cs = 2.04. ࠷΋஗͍Ի଎͕ܭࢉྖҬΛ
    ԣ੾Δ࣌ؒ໿ 1.0 ͷ 8 ഒͰ͋Δ t = 8.0 ·Ͱܭࢉ
    • ॳظ৚݅Ͱ͸ѹྗ͸શཻࢠҰఆͰ଎౓ 0 ີ౓ͷෆ࿈ଓ෦෼͕ଘࡏ.
    ▶ Կ΋ى͜Βͳ͍ͷ͕ਖ਼ղ
    • ද໘ுྗͷޮՌͰؙ͘ͳͬͯ͠·͏
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 11 / 45

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  12. ֤छεΩʔϜʹ͍ͭͯ Standard SPH ๏
    ਓ޻೪ੑ
    ඞཁੑͱ໰୊఺
    • ඞཁੑ
    ▶ ඇ೪ੑѹॖੑྲྀମͰ͸িܸ೾͕ൃੜɻিܸ೾෦෼Ͱ͸෺ཧྔ͸ෆ࿈ଓʹมԽ. ͦΕΛ਺஋ܭ
    ࢉͰଊ͑ΒΕΔΑ͏ʹ͢ΔͨΊʹਓ޻೪ੑͰ׈Β͔ʹ͢Δ
    • ໰୊఺
    ▶ ਓ޻೪ੑʹ͸ύϥϝʔλ͕ଘࡏ. ύϥϝʔλͷ஋ʹΑͬͯγϛϡϨʔγϣϯ݁Ռ͕มԽ
    ▶ িܸ೾ྖҬҎ֎Ͱ΋ਓ޻೪ੑ͕༗ޮʹͳΔ (γΞྖҬͳͲ)ɻͨͩ͠γΞྖҬͰͷਓ޻೪ੑ
    Λ཈͑Δॲํᝦ͸ଘࡏ (Balsara switch [Balsara, 1995])
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 12 / 45

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  13. ֤छεΩʔϜʹ͍ͭͯ Standard SPH ๏
    ఺ݯരൃ
    ڧ͍িܸ೾͕ൃੜ͢Δ 3 ࣍ݩͷ໰୊
    0 < x, y, z < 1 ͷྖҬͷਅΜதʹ߹ܭ 1 ͷΤω
    ϧΪʔΛׂΓৼΔɻີ౓͸ 1ɺ଎౓͸ 0. ֎ଆͷ
    ΤωϧΪʔ͸΄΅ 0
    ಺෦ΤωϧΪʔϓϩϑΝΠϧ
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 13 / 45

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  14. ֤छεΩʔϜʹ͍ͭͯ Standard SPH ๏
    SSPH
    • ਓ޻೪ੑ܎਺খ͍͞ͱϊ
    Πζ͕େ͖͘ͳΔɻେ͖
    ͗͢Δͱղ͕ಷΔɻ
    • ͓͓ΉͶղੳղͱҰக͠
    ͍ͯΔ͕ɺਓ޻೪ੑڧ͘
    ͯ͠΋଎౓Ͱৼಈ͕ൃੜ
    • ѹྗʹ΋ৼಈ͕͋Δ
    • ௿ີ౓ྖҬͷѹྗʹେ͖
    ͳޡࠩ
    • িܸ೾ޙ໘Ͱͷີ౓͕ղ
    ੳղͱඍົʹҰக͍ͯ͠
    ͳ͍
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 14 / 45

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  15. ֤छεΩʔϜʹ͍ͭͯ Standard SPH ๏
    ۭؒθϩ࣍ޡࠩ
    • ཭ࢄԽͷࡍʹཻࢠ෼෍ͷඇ౳ํੑ͔Β͘Δۭؒθϩ࣍ͷޡ͕ࠩൃੜ
    F(r) =

    F(r′
    )W(|r − r′
    |, h)d3r′
    + O(h2)
    =

    j
    mj
    ρj
    FjW(|r − rj|, h) + O(?) + O(h2)
    (10)
    Fj Λ ri
    ͷपΓͰల։͢ΔͱɺFj ∼ Fi − ∇iFi · rij + O(h2) ͱͳΔɻ͜ΕΛ F(ri) = Fi ʹ୅
    ೖ͢Δͱ
    Fi ∼ FiΣj
    mj
    ρj
    Wij(h) − ∇iFi · Σj
    mj
    ρj
    rijWij(h) + O(h2) (11)
    ࣜ (11) ͕ O(h2)(཭ࢄԽલͷޡࠩͱಉ͡) ʹͳΔʹ͸ɺ
    Σj
    mj
    ρj
    Wij(h) = 1, Σj
    mj
    ρj
    rijWij(h) = 0 (12)
    ͱͳΔඞཁ͕͋Δ͕ɺ࣮ࡍͷܭࢉͰ͸ඞͣ͠΋͜͏ͳΔͱ͸ݶΒͳ͍ɻಉ༷ͳٞ࿦͔Βɺ
    SPH ͷӡಈํఔࣜɺΤωϧΪʔํఔࣜ΋ۭؒθϩ࣍ͷޡ͕ࠩଘࡏ͢Δ͜ͱ͕ಋ͖ग़͞ΕΔɻ
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 15 / 45

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  16. ֤छεΩʔϜʹ͍ͭͯ SSPH with Artificial Conductivity ๏
    SSPH with ArtCond ๏
    • [Price, 2008] ͕ߟҊ
    • ΤωϧΪʔ֦ࢄ߲ΛೖΕɺ಺෦ΤωϧΪʔΛ׈Β͔ʹ͢Δ͜ͱͰѹྗδϟϯϓΛղফ͢
    Δ͜ͱΛ໨ࢦ͍ͯ͠Δ
    • ಋग़ͷࡍʹີ౓ͷۭؒඍ෼͸ೖͬͨ··
    SSPH with ArtCond ಋग़ʹ༻͍ΔѹॖੑྲྀମͷӡಈํఔࣜɺΤωϧΪʔํఔࣜ
    dv(r)
    dt
    = −
    1
    ρ(r)
    ∇P(r) + FAV (αAV ) (13)
    du(r)
    dt
    = −
    P(r)
    ρ(r)
    ∇ · v(r) + GAV (αAV ) + HAC(αu) (14)
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 16 / 45

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  17. ֤छεΩʔϜʹ͍ͭͯ SSPH with Artificial Conductivity ๏
    SSPH with Artificial Conductivity
    িܸ೾؅໰୊ ڧ͍িܸ೾ (ॳظ৚݅Ͱ 104 ͷѹྗࠩ)
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 17 / 45

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  18. ֤छεΩʔϜʹ͍ͭͯ SSPH with Artificial Conductivity ๏
    • ਓ޻೤఻ಋ͕͋ͬͯ΋ܗ͸େ͖͘มܗ
    • ೤఻ಋͷޮՌͰෆ࿈ଓ෦෼͕΅΍͚Δ
    • ਓ޻೤఻ಋ͸ɺࠓߟ͍͑ͯΔ෺ཧ (ΦΠϥʔํఔࣜ) ʹ͸ೖ͍ͬͯͳ͍ޮՌΛೖΕͯ͠
    ·͍ͬͯΔ
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 18 / 45

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  19. ֤छεΩʔϜʹ͍ͭͯ Density-Independent SPH ๏
    DISPH ๏
    • [Saitoh and Makino, 2013] ʹΑͬͯߟҊ͞Εͨख๏
    • ઀৮ෆ࿈ଓ໘Ͱ࿈ଓͱͳΔѹྗΛ༻ཻ͍ͯࢠͷମੵཁૉΛۙࣅ͠ɺѹྗͷۭؒඍ෼Λ
    ࢖༻
    ▶ ઀৮ෆ࿈ଓ໘ͷѻ͍͕ྑ͘ͳΔ
    SPH ͷ෺ཧྔ
    F(ri) ≈

    F(r′
    )W(|ri − r′
    |, h)d3r′


    j
    mj
    ρj
    FjW(|ri − rj|, h)
    (15)
    ρi =

    j
    mjWij(h) (16)
    ಋग़தͰ ∇ρ Λ࢖༻
    DISPH ͷ෺ཧྔ
    F(ri) ≈

    F(r′
    )W(|ri − r′
    |, h)d3r′


    j
    Uj
    qj
    FjW(|ri − rj|, h)
    (17)
    qi =

    j
    UjWij(h) =
    Pi
    (γ − 1)
    (18)
    ಋग़தͰ ∇ρ ͸࢖Θͣɺ∇q Λ࢖༻. ີ౓
    ʹཅʹґଘ͠ͳ͍
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 19 / 45

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  20. ֤छεΩʔϜʹ͍ͭͯ Density-Independent SPH ๏
    DISPH
    িܸ೾؅໰୊ ڧ͍িܸ೾ (ॳظ৚݅Ͱ 104 ͷѹྗࠩ)
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 20 / 45

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  21. ֤छεΩʔϜʹ͍ͭͯ Density-Independent SPH ๏
    • ද໘ுྗͷޮՌ͕ͳ͘ͳ͍ͬͯΔ
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 21 / 45

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  22. ֤छεΩʔϜʹ͍ͭͯ Density-Independent SPH ๏
    DISPH
    • DISPH ͸ڧ͍িܸ೾ʹର
    ͯ͠ෆಘҙͳ͸͕ͣͩɺ
    ద੾ͳ೪ੑ͕͋Ε͹े෼
    ͳੑೳ
    • ద੾ͳ೪ੑͰ͋Ε͹ɺج
    ຊతʹ SPH ͱࣅͨΑ͏
    ͳ݁Ռ
    • SPH ͱͷҧ͍ͱͯ͠͸௿
    ີ౓ྖҬͰͷѹྗͷޡࠩ
    ͕ൺֱత཈͑ΒΕ͍ͯΔ
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 22 / 45

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  23. ֤छεΩʔϜʹ͍ͭͯ Godunov SPH ๏
    Godunov SPH ๏
    • [Inutsuka, 2002] ʹΑͬͯߟҊ͞Εͨख๏ɻ
    • ཻࢠͷ૬ޓ࡞༻ܭࢉͷࡍʹ Riemann solver Λ࢖༻
    • ͨͩ͠௨ৗͷ SPH ΑΓ΋ܭࢉίετߴ, ίϯύΫτͰ͸ͳ͍Ψ΢γΞϯΧʔωϧΛ࢖͏
    ඞཁੑ͋Γ
    ▶ ຊൃදͰ͸ [Cha and Whitworth, 2003] ʹΑͬͯఏҊ͞Ε͍ͯΔ GSPH Case3 Λ࢖༻
    GSPH Case3 ͷӡಈํఔࣜ, ΤωϧΪʔํఔࣜ
    dvi
    dt
    = −Σjmjp∗
    ij
    [
    1
    ρ2
    i
    ∇iWij(hi) +
    1
    ρ2
    j
    ∇iWij(hj)
    ]
    (19)
    dui
    dt
    = −Σjmjp∗
    ij
    (v∗
    ij
    − vi) ·
    [
    1
    ρ2
    i
    ∇iWij(hi) +
    1
    ρ2
    j
    ∇iWij(hj)
    ]
    (20)
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 23 / 45

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  24. ֤छεΩʔϜʹ͍ͭͯ Godunov SPH ๏
    ͋Δ෺ཧྔ F ʹରͯ͠ɺͦͷॏΈ͖ͭฏۉΛ F∗
    ij
    Λ

    F(r)
    ρ2(r)
    W(|r − ri|, h(r))W(|r − rj|, h(r))d3r
    = F∗
    ij

    1
    ρ2(r)
    W(|r − ri|, h(r))W(|r − rj|, h(r))d3r
    (21)
    Λຬͨ͢Α͏ʹ༩͑ΒΕΔͱ͢Δɻ͜ΕΛྑ͍ਫ਼౓Ͱຬͨ͢Α͏ͳ F∗
    ij
    Λ༻͍Δɻ
    GSPH Case3-2 ͷӡಈํఔࣜ, ΤωϧΪʔํఔࣜ
    dvi
    dt
    = −Σjmjp∗
    ij
    [
    1
    ρ2
    i
    ∇iWij(hi) +
    1
    ρ2
    j
    ∇iWij(hj)
    ]
    (22)
    dui
    dt
    = −Σjmjp∗
    ij
    (
    vi + vj
    2
    − vi) ·
    [
    1
    ρ2
    i
    ∇iWij(hi) +
    1
    ρ2
    j
    ∇iWij(hj)
    ]
    (23)
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 24 / 45

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  25. ֤छεΩʔϜʹ͍ͭͯ Godunov SPH ๏
    Riemann Solver ʹ͍ͭͯ
    • ॳظ৚݅
    W = (ρ, P, v)
    • ղͷλΠϓ
    P∗
    L
    = P∗
    R
    , v∗
    L
    = v∗
    R
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 25 / 45

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  26. ֤छεΩʔϜʹ͍ͭͯ Godunov SPH ๏
    • 1 ࣍ݩ Riemann solver ͷೖྗ஋ͷܾΊํ͸༷ʑɻԿΛग़ྗ஋ʹ͢Δ͔༷ʑɻGodunov ๏
    (Grid ๏) Ͱ༻͍ΒΕ͍ͯΔ MUSCL ๏Λ૊ΈࠐΜͩΓ΋Ͱ͖Δ
    • ຊൃදͰ͸ɺGSPH ಺Ͱ༻͍Δ Riemann solver ͷೖྗ஋ͱཻͯ͠ࢠ i ͱཻࢠ j ͷ෺ཧྔ
    Λɺग़ྗ஋ͱͯ͠ star region Ͱͷ஋Λ༻͍Δɻ
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 26 / 45

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  27. ֤छεΩʔϜʹ͍ͭͯ Godunov SPH ๏
    িܸ೾؅໰୊
    GSPH Case3 GSPH Case3-2 (v∗
    ij
    = vi+vj
    2
    )
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 27 / 45

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  28. ֤छεΩʔϜʹ͍ͭͯ Godunov SPH ๏
    ڧ͍িܸ೾ (ॳظ৚݅Ͱ 104 ͷѹྗࠩ)
    GSPH Case3 GSPH Case3-2 (v∗
    ij
    = vi+vj
    2
    )
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 28 / 45

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  29. ֤छεΩʔϜʹ͍ͭͯ Godunov SPH ๏
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 29 / 45

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  30. ֤छεΩʔϜʹ͍ͭͯ Godunov SPH ๏
    GSPH Case3
    • ύϥϝʔλͳ͠ͰɺSPH,DISPH ͰݟΒΕ
    ͨিܸ೾ޙ໘Ͱͷ଎౓ͷৼಈ΍ີ౓ͷղ
    ੳղͱͷͣΕ͕ͳ͘ͳ͍ͬͯΔ
    • ௿ີ౓ྖҬͷѹྗ͸ SPH ͱಉ༷ޡ͕ࠩ
    ͋Δ
    • ௿ີ౓ྖҬͷ଎౓ʹൺֱతେ͖ͳޡࠩ
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 30 / 45

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  31. ֤छεΩʔϜʹ͍ͭͯ ·ͱΊ
    ·ͱΊ
    • DISPH ͸઀৮ෆ࿈ଓ໘ͷѻ͍͕ྑ͍ (઀৮ෆ࿈ଓ໘ͷͨΊͷ௥Ճͷࢄҳ߲ɺύϥϝʔλ
    ͳ͠)
    • ద੾ͳਓ޻೪ੑύϥϝʔλΛબ୒͢Ε͹ɺڧ͍িܸ೾΍ٸܹͳѹྗޯ഑ʹରͯ͠΋े෼
    ͳੑೳ
    • GSPH ͸িܸ೾ΛҰ੾ͷύϥϝʔλͳ͠Ͱѻ͑Δ (ѹྗʹ Riemann solver ͷղΛ༻͍Δ
    ͜ͱͰɺద੾ͳ೪ੑ͕෇Ճ͞ΕΔ)
    ▶ Riemann Solver Λ DISPH ʹ૊ΈࠐΉ (ѹྗ෦෼Λ Riemann solver ͷղΛ༻͍Δ) ͜ͱͰɺ
    ύϥϝʔλͳ͠Ͱ઀৮ෆ࿈ଓ໘, িܸ೾Λѻ͑ΔεΩʔϜΛ࡞Δ
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 31 / 45

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  32. Godunov DISPH ๏ಋग़
    Godunov DISPH ๏ಋग़ (೤ྗֶୈҰ๏ଇΛى఺)
    ཻࢠ i ͷඍখ಺෦ΤωϧΪʔมԽ͸ɺཻࢠ͕֎෦͔Βड͚ΔମੵมԽΛ௨ͨ͡࢓ࣄʹΑΔ΋
    ͷ (ॏ৺ͷҐஔมԽΛ௨ͨ͡࢓ࣄ͸ӡಈΤωϧΪʔʹ)
    dUi = WV olume
    i
    (24)
    DISPH Ͱ͸ (SPH Ͱ΋)
    WV olume
    i
    = −PidVi (25)
    ཻࢠ i ͕ dt ͷؒʹ֎෦͔Βड͚Δѹྗ͸ɺPi + ϵ Ͱ͋ΔͱԾఆ͠ɺೋ࣍ͷඍখྔΛແࢹͯ͠
    ͍Δ. (ྲྀମํఔࣜͷΤωϧΪʔͷࣜͰ΋ಉ༷) ແݶݸͷཻࢠ, ແݶখͷ smoothing length, ແ
    ݶখͷλΠϜεςοϓΛ࢖͑͹ਖ਼͍͠͸ͣ
    ৽͍͠ߟ͑ํ
    Pix Λɺཻࢠ i Λத৺ͱͨ͋͠ΒΏΔํ޲͔Βཻࢠ i ͕ड͚Δѹྗͷ͋Δछͷۭ࣌ؒؒฏۉྔ
    ͱͯ͠
    WV olume
    i
    = −PixdVi (26)
    ͕੒Γཱͭͱ͢Δɻ
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 32 / 45

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  33. Godunov DISPH ๏ಋग़
    ཻࢠ i ͷମੵཁૉ
    Vi =
    Ui
    qi
    (27)
    ཻࢠ i ͷӨڹ൒ܘ಺ͷཻࢠ਺ΛҰఆʹ͢Δͱ͍͏৚݅
    qi
    Ui
    hD
    i
    = const (28)
    ΤωϧΪʔํఔࣜ͸
    dUi
    dt
    = fgrad
    i
    N

    j
    PixUiUj
    q2
    i
    vij · ∇iWij(hi). (29)
    fgrad
    i
    =

    1 +
    hi
    Dqi
    N

    j
    Uj
    ∂Wij(hi)
    ∂hi


    −1
    . (30)
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 33 / 45

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  34. Godunov DISPH ๏ಋग़
    Pix ͷఆٛ
    Pix ΛɺP∗
    ij
    Λཻࢠ i ͱཻࢠ j ͷ෺ཧྔΛೖྗ஋ͱͨ͠ࡍͷ Star region Ͱͷѹྗͷ஋ͱͯ͠
    Pix
    N

    j
    UiUj
    q2
    i
    vij · ∇iWij(hi) =
    N

    j
    P∗
    ij
    UiUj
    q2
    i
    vij · ∇iWij(hi), (31)
    ͕੒Γཱͭͱఆٛ͢Δɻ
    • P∗
    ij
    Λཻࢠ i ཻ͕ࢠ j ͔Βड͚Δѹྗͷ࣌ؒฏۉͱ͠ɺۭؒฏۉ͸೚ҙੑ͕ଘࡏ͢Δɻ
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 34 / 45

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  35. Godunov DISPH ๏ಋग़
    Godunov DISPH ͷӡಈํఔࣜ, ΤωϧΪʔํఔࣜ
    ΤωϧΪʔํఔࣜ͸
    dUi
    dt
    = fgrad
    i
    N

    j
    P∗
    ij
    UiUj
    q2
    i
    vij · ∇iWij(hi). (32)
    ӡಈํఔࣜ͸ɺ࡞༻൓࡞༻Λຬͨ͠ΤωϧΪʔอଘ΋ຬͨ͞ͳ͚Ε͹ͳΒͳ͍ͱ͍͏৚͔݅
    ΒٻΊΒΕΔɻ
    mi
    dvi
    dt
    = −
    N

    j
    [
    fgrad
    i
    P∗
    ij
    UiUj
    q2
    i
    ∇iWij(hi) + fgrad
    j
    P∗
    ij
    UiUj
    q2
    j
    ∇iWij(hj)
    ]
    (33)
    Pix = Pi ͷͱ͖ (DISPH Ͱ࢖༻͞Ε͍ͯΔ) ͸, γάϚͷத਎ͷୈҰ߲ͷѹྗ͸ Piɺୈೋ߲ͷ
    ѹྗ͸ Pj
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 35 / 45

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  36. ςετܭࢉ Riemann ໰୊
    Godunov DISPH
    িܸ೾؅໰୊ ڧ͍িܸ೾ (ॳظ৚݅Ͱ 104 ͷѹྗࠩ)
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 36 / 45

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  37. ςετܭࢉ 2D ѹྗฏߧ
    • ද໘ுྗʹΑΔޮՌ͸ݟΒΕͳ͍
    • ෺ཧతͳղͱͳ͍ͬͯΔ
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 37 / 45

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  38. ςετܭࢉ ఺ݯരൃ
    GDISPH • GSPH ͱಉ༷ʹ,SPH,DISPH ͱൺֱ͠
    ͯύϥϝʔλͳ͠Ͱ଎౓ͷৼಈΛ཈͑Β
    Ε͍ͯΔ
    • SPH,DISPH ͱҧͬͯ, িܸ೾ޙ໘Ͱͷີ
    ౓ͷ஋΋ύϥϝʔλͳ͠ͰղੳղͱҰக
    • SPH,GSPH ͱҧͬͯ, ௿ີ౓ྖҬͷѹྗ
    ޡ͕ࠩ཈͑ΒΕ͍ͯΔ
    • ௿ີ౓ྖҬͰ଎౓ɺ಺෦ΤωϧΪʔʹৼ
    ಈ͕ൃੜ
    GDISPH ͸িܸ೾ޙ໘ͷੑೳ͸ GSPH ͱಉ༷
    (SPH,DISPH ʹൺ΂ͯྑ͍݁Ռ). ௿ີ౓ྖҬ
    Ͱ͸ GDISPH ݻ༗ͷ໰୊͕ൃੜɻͨͩ͠ɺଞ
    ͷεΩʔϜ΋௿ີ౓ྖҬͰݻ༗ͷ໰୊͋Γɻ
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 38 / 45

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  39. ςετܭࢉ 2D Kelvin-Helmholtz ෆ҆ఆੑ
    2D Kelvin-Helmholtz ෆ҆ఆੑ
    Ґஔ y = 0.25, y = 0.75 ͷͦΕͧΕʹɺy ࣠ํ޲ͷ଎౓ʹ 2 ೾௕෼ͷઁಈΛ༩͍͑ͯΔ
    ෆ҆ఆੑͷ੒௕ͷλΠϜεέʔϧ τKH = 1.06 ͷ໿ 4 ഒͷ t = 4.0 ·Ͱܭࢉ
    Balsara switch Λ෇͚ͯܭࢉ
    ॳظ৚݅ ༩͑ͨઁಈ
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 39 / 45

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  40. ςετܭࢉ 2D Kelvin-Helmholtz ෆ҆ఆੑ
    Balsara switch ͱ͸
    • [Balsara, 1995] ʹΑͬͯఏҊ͞Εͨख๏
    • ਓ޻೪ੑ߲͸ຊདྷিܸ೾ྖҬͰͷΈಇ͍ͯ΄͍͕͠ɺγΞྖҬͰ΋ಇ͍ͯ͠·͏ɻ
    • ͦΕΛ๷͙ͨΊɺBalsara switch ͕Α͘࢖ΘΕ͍ͯΔɻ
    Balsara switch
    ਓ޻೪ੑ߲ʹ 0.5(Fi + Fj) Λ͔͚ΔɻγΞྖҬͰ͸ Fi, Fj Λ 1 ΑΓখ͘͞ɺিܸ೾ྖҬͰ͸
    Fi, Fj ͕ 1 ʹͳΔΑ͏ʹઃఆɻ
    Fi =
    |∇i · vi|
    |∇i · vi| + |∇i × vi| + 0.0001ci/hi
    (34)
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 40 / 45

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  41. ςετܭࢉ 2D Kelvin-Helmholtz ෆ҆ఆੑ
    GDISPH ΁ͷ Balsara switch ͷద༻
    • GSPHɺGDISPH ʹ Balsara switch ͷద༻Λߦͬͨྫ͸ͳ͍ɻ
    • DISPH ͸ (ඇਓ޻೪ੑ߲)+(ਓ޻೪ੑ߲) ʹཅʹ෼͔Ε͍ͯΔ͕ɺGDISPH ͸ 1 ͭͷ߲ʹ
    ·ͱ·͍ͬͯͯ෼཭͍ͯ͠ͳ͍ɻ
    • ཧ૝͸ GDISPH தͷ P∗
    ij
    Λ (ඇ೪ੑ߲)+(೪ੑ߲) ʹཅʹ෼͚Δ͜ͱ
    զʑͷఏҊख๏
    • (DISPH ͷඇਓ޻೪ੑ߲)+0.5(Fi + Fj)(GDISPH ͷ߲ - DISPH ͷඇਓ޻೪ੑ߲) Λɺ
    GDISPH ͷࣜͱͯ͠༻͍Δɻ
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 41 / 45

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  42. ςετܭࢉ 2D Kelvin-Helmholtz ෆ҆ఆੑ
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 42 / 45

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  43. ςετܭࢉ 2D Kelvin-Helmholtz ෆ҆ఆੑ
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 43 / 45

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  44. ςετܭࢉ 2D Kelvin-Helmholtz ෆ҆ఆੑ
    ݪҼͷ༧૝
    ϊΠζͷਖ਼ମ
    ཻ֤ࢠʹ͓͚Δɺۭؒ 0 ࣍ͷޡࠩ΍
    O(h2) ͷޡࠩͷ͔͔Γํͷҧ͍ʹΑΔ΋
    ͷɻཻ֤ࢠؒͷඍົͳ਺஋ͷͣΕ͕ɺ
    ඇৗʹ೾௕ͷখ͍͞ઁಈͱͳΔɻ
    ϊΠζ͕େ͖͘੒௕͢Δཧ༝
    • balsara switch ͷಋೖʹΑͬͯγΞྖ
    ҬͰͷ೪ੑ͕΄΅ 0 ʹͳΔɻ
    • ڪΒ͘෺ཧతʹ͸ɺྲྀମͷ೪ੑ͕େ
    ͖͘ͳΔͱɺKH ෆ҆ఆੑ͕ى͜Δ
    ࠷খͷ೾௕΋େ͖͘ͳΔɻ
    ೪ੑ͕ 0 ͳΒ͹ɺ
    ͲͷΑ͏ͳ೾௕ʹରͯ͠΋ෆ҆ఆੑ੒௕
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 44 / 45

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  45. ݁࿦
    ݁࿦
    • Godunov DISPH
    ▶ DISPH ʹ Riemann solver Λ૊ΈࠐΜͩ Godunov DISPH Λ࡞Δ͜ͱͰɺ઀৮ෆ࿈ଓ໘΋ি
    ܸ೾΋ύϥϝʔλͳ͠Ͱѻ͑ΔΑ͏ʹͳͬͨ
    ▶ Godunov DISPH ΋௨ৗͷ SPH ͱಉ༷ɺγΞྖҬͰ΋೪ੑΛ͔͚ͯ͠·͏໰୊఺͕͋Δɻ
    ͦͷ໰୊఺ղফͷͨΊɺGDISPH ʹ Balsara switch Λ૊ΈࠐΉͨΊͷख๏ΛߟҊɻKH ෆ҆
    ఆੑͷܭࢉͰ DISPH ͱಉ༷ͳ݁Ռ
    • Kelvin-Helmholtz ෆ҆ఆੑ
    ▶ ॳظ৚݅ʹΑͬͯ͸ϊΠζʹΑΔઁಈ΋੒௕͢Δ
    ▶ ͲͷΑ͏ͳঢ়گͰ΋ɺཻࢠؒڑ཭ఔ౓ͷ೾௕ͷઁಈͷ੒௕Λ཈͑ΒΕΔํ๏Λݟ͚͍ͭͨɻ
    ࠓޙͷ՝୊
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 45 / 45

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  46. ݁࿦
    ग़య
    Figure: ”Fancy SPH convolution scheme (verbose, modified colors scheme)” created by Jlcercos is
    licenced under CC BY-SA 4.0(https://creativecommons.org/licenses/by-sa/4.0/)
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 45 / 45

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  47. ݁࿦
    Reference I
    [Balsara, 1995] Balsara, D. S. (1995).
    Von neumann stability analysis of smoothed particle hydrodynamicsŠsuggestions for optimal algorithms.
    Journal of Computational Physics, 121(2):357–372.
    [Brookshaw, 1985] Brookshaw, L. (1985).
    A method of calculating radiative heat diffusion in particle simulations.
    Publications of the Astronomical Society of Australia, 6(2):207 r 210.
    [Cha and Whitworth, 2003] Cha, S.-H. and Whitworth, A. P. (2003).
    Implementations and tests of godunov-type particle hydrodynamics.
    Monthly Notices of the Royal Astronomical Society, 340(1):73–90.
    [Garc´
    ıa-Senz et al., 2012] Garc´
    ıa-Senz, D., Cabez´
    on, R. M., and Escart´
    ın, J. A. (2012).
    Improving smoothed particle hydrodynamics with an integral approach to calculating gradients.
    Astronomy & astrophysics, 538:A9.
    [Inutsuka, 2002] Inutsuka, S.-i. (2002).
    Reformulation of smoothed particle hydrodynamics with riemann solver.
    Journal of Computational Physics, 179.
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 45 / 45

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  48. ݁࿦
    Reference II
    [Price, 2008] Price, D. J. (2008).
    Modelling discontinuities and kelvin–helmholtz instabilities in sph.
    Journal of Computational Physics, 227(24):10040–10057.
    [Saitoh and Makino, 2013] Saitoh, T. R. and Makino, J. (2013).
    A DENSITY-INDEPENDENT FORMULATION OF SMOOTHED PARTICLE HYDRODYNAMICS.
    The Astrophysical Journal, 768(1):44.
    [Wadsley et al., 2017] Wadsley, J. W., Keller, B. W., and Quinn, T. R. (2017).
    Gasoline2: a modern smoothed particle hydrodynamics code.
    Monthly Notices of the Royal Astronomical Society, 471(2):2357–2369.
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 45 / 45

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  49. ݁࿦
    SPH GDF ๏
    • Geometric Density Average Force (GDF)([Wadsley et al., 2017])
    • Gasoline2 Ͱ༻͍ΒΕ͍ͯΔ͓ΓɺSPH ๏ΑΓ΋઀৮ෆ࿈ଓ໘ͷѻ͍͕ྑ͍
    GDF ๏ͷӡಈํఔࣜಋग़
    • ྲྀମͷӡಈํఔࣜΛมܗ͢Δ

    ∇P
    ρ
    = −
    1
    ρ2−σ

    (
    P
    ρσ−1
    )

    P
    ρσ
    ∇ρσ−1 (35)
    • σ = 1 ͷͱ͖ (GDF ಋग़Ͱ༻͍Δ, ີ౓ͷۭؒඍ෼͕ͳ͍)

    ∇P
    ρ
    = −
    1
    ρ
    ∇P −
    P
    ρ
    ∇1 (36)
    • σ = 2 ͷͱ͖ (௨ৗͷ SPH Ͱ࢖༻, ີ౓ͷۭؒඍ෼͕͋Δ)

    ∇P
    ρ
    = −∇
    (
    P
    ρ
    )

    P
    ρ2
    ∇ρ (37)
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 45 / 45

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  50. ݁࿦
    • ͜ΕΛ F(r) =

    j
    mj
    ρj
    FjW (|r − rj|, h(r)) Ͱ཭ࢄԽ͢Δ (ࣜ (3)) ͱ
    dvi
    dt
    = −

    j
    mj
    (
    Pi
    ρσ
    i
    ρ2−σ
    j
    +
    Pj
    ρ2−σ
    i
    ρσ
    j
    )
    ∇iWij(h) (38)
    • σ = 1 Λ୅ೖ (Α͘༻͍ΒΕ͍ͯΔ standard SPH ͸ σ = 2 ͷ࣌ͱಉ༷ͷܗ)
    dvi
    dt
    = −

    j
    mj
    (
    Pi + Pj
    ρiρj
    )
    ∇iWij(h) (39)
    ΤωϧΪʔํఔࣜ΋ಉ༷ͷܗͷ΋ͷΛ࢖༻
    dui
    dt
    =

    j
    mj
    (
    Pi
    ρiρj
    )
    vij · ∇iWij(h) (40)
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 45 / 45

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  51. ݁࿦
    িܸ೾؅໰୊
    ڧ͍িܸ೾ (ॳظ৚݅Ͱ 104 ͷѹྗࠩ)
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 45 / 45

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  52. ݁࿦
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 45 / 45

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  53. ݁࿦
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 45 / 45

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  54. ݁࿦
    γΞ଎౓Λ׈Β͔ʹભҠͤ͞Δ
    R(y) =
    1
    1 + exp [−2(y − 0.25)/0.025]
    1
    1 + exp [2(y − 0.75)/0.025]
    (41)
    vx(y) = vx,l + R(y)[vx,h − vx,l] (42)
    ॳظ৚݅
    x ࣠ํ޲ͷ଎౓
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 45 / 45

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  55. ݁࿦
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 45 / 45

    View Slide

  56. ݁࿦
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 45 / 45

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  57. ݁࿦
    վળࡦ
    • ͲͷΑ͏ͳঢ়گʹ͓͍ͯ΋ɺ෺ཧతͳઁಈʹΑΔෆ҆ఆੑͷ੒௕͸ڐ͠ɺ਺஋తͳϊΠ
    ζʹΑΔઁಈͷ੒௕͸཈͑Δɻ
    → balsara switch ͷڧ͞Λௐઅ͢Δ͜ͱͰ࣮ݱΛ໨ࢦ͢ɻ
    Balsara switch
    • ਓ޻೪ੑ߲ʹ 0.5(Fi + Fj) Λ͔͚Δɻ
    Fi =
    |∇i · vi|
    |∇i · vi| + |∇i × vi| + 0.0001ci/hi
    (43)
    զʑͷఏҊख๏
    • Fi, Fj ͷ୅ΘΓʹ F′
    i
    , F′
    j
    Λ༻͍Δɻ
    • β ͸ [0, 1] ͷ࣮਺ΛऔΔ೚ҙύϥϝʔλɻβ = 1 Ͱ balsara switch Φϯɹ β = 0 Ͱ balsara
    switch Φϑ
    F′
    i
    = 1 + β(Fi − 1) (44)
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 45 / 45

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  58. ݁࿦
    ೪ੑΛ༻͍Δཧ༝
    • ΤωϧΪʔ֦ࢄ߲͸ີ౓͕ಉ͡ʹͳΔΑ͏ಇ͖ɺ೪ੑ߲͸଎౓͕ಉ͡ʹͳΔΑ͏ʹಇ
    ͘ɻγΞྖҬͰ͸ͳΔ΂͘ಇ͔ͳ͍Α͏ʹ͍ͨ͠ɻ
    ▶ গͳ͍֦ࢄ or ೪ੑͰɺཻࢠؒڑ཭ఔ౓ͷ೾௕ͷઁಈͷ੒௕Λ཈͍͑ͨ
    ▶ ΤωϧΪʔ֦ࢄ߲෇͖ or ೪ੑ߲෇͖ͷྲྀମํఔࣜͷઢܗղੳΛߦ͏͜ͱͰௐ΂͍ͨɻࠓޙ
    ͷ՝୊
    • KH ෆ҆ఆੑͷλΠϜεέʔϧΛݟΔͱɺີ౓ͷۉ࣭ԽΑΓ΋γΞ଎౓ͷۉ࣭Խͷ΄͏
    ͕ KH ෆ҆ఆੑʹ༩͑ΔӨڹ͕ڧͦ͏
    ▶ ೪ੑͷ΄͏͕ΑΓޮՌతʹϊΠζͷ੒௕Λ཈͑ΒΕΔͱ༧૝Ͱ͖Δ
    τkh =
    λ(ρh + ρl)

    ρhρl|vx,h − vx,l|
    (45)
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 45 / 45

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  59. ݁࿦
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 45 / 45

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  60. ݁࿦
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 45 / 45

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  61. ݁࿦
    Riemann ໰୊
    {
    ρ = 1, P = 0.4, v = −2 x < 0 ͷͱ͖
    ρ = 1, P = 0.4, v = 2 x > 0 ͷͱ͖
    (46)
    ີ౓ ѹྗ ଎౓
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 45 / 45

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  62. ݁࿦
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 45 / 45

    View Slide

  63. ݁࿦
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 45 / 45

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