Formation of Astrophysical Objects 2022

Transcript

1. Riemann solverΛ༻͍ͨDensity-Independent Smoothed
   Particle Hydrodynamics(DISPH)ͷ࠶ߏ੒
   ᴷ ৽ SPH εΩʔϜ:Godunov DISPH ๏ʹ͍ͭͯ ᴷ
   ౬ઙ୓޺, ৿ਖ਼෉
   ஜ೾େֶ
   December 2, 2022
   ౬ઙ ୓޺, ৿ ਖ਼෉ (ஜ೾େֶ) DISPH, Godunov SPH, Godunov DISPH, IA December 2, 2022 1 / 22
   

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2. ໨࣍
   1 ֤छεΩʔϜʹ͍ͭͯ
   SPH ๏
   Density-Independent SPH ๏,
   GSPH ๏
   Godunov DISPH ๏
   Integral Approach
   2 ςετܭࢉ
   ࣮ݧͷ໨త
   Riemann ໰୊
   2D ੩ਫѹฏߧ
   2D Kelvin-Helmholtz ෆ҆ఆੑ
   ·ͱΊ
   3 sheer switch ͷվྑʹ޲͚ͯ
   ໰୊఺
   ݪҼͷ༧૝
   վળࡦ
   4 ࠓޙͷํ਑
   ౬ઙ ୓޺, ৿ ਖ਼෉ (ஜ೾େֶ) DISPH, Godunov SPH, Godunov DISPH, IA December 2, 2022 2 / 22
   

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3. ֤छεΩʔϜʹ͍ͭͯ SPH ๏
   SPH ๏ಋग़ʹ༻͍ΔѹॖੑྲྀମͷӡಈํఔࣜɺΤωϧΪʔํఔࣜ
   dv(r)
   dt
   = −
   1
   ρ(r)
   ∇P(r) + FAV (α) (1)
   du(r)
   dt
   = −
   P(r)
   ρ(r)
   ∇ · v(r) + GAV (α) (2)
   • σϝϦοτ
   ▶ ྲྀମͷ઀৮ෆ࿈ଓ໘Λ͏·͘ѻ͑ͣɺͦͷͨΊྲྀମͷෆ҆ఆੑͷ੒௕Λ๦͛Δ
   ▶ িܸ೾Λଊ͑ΔͨΊʹਓ޻తͳࢄҳ߲͕ӡಈํఔࣜɺΤωϧΪʔํఔࣜʹඞཁɻ
   ਓޱ೪ੑͷڧ͞Λௐઅ͢Δ೚ҙύϥϝʔλ α ΛਓؒͷखͰௐ੔͢Δඞཁ
   ▶ ཭ࢄԽͷࡍʹཻࢠ෼෍ͷඇ౳ํੑ͔Β͘Δۭؒθϩ࣍ͷޡ͕ࠩൃੜ
   ౬ઙ ୓޺, ৿ ਖ਼෉ (ஜ೾େֶ) DISPH, Godunov SPH, Godunov DISPH, IA December 2, 2022 3 / 22
   

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4. ֤छεΩʔϜʹ͍ͭͯ Density-Independent SPH ๏, GSPH ๏
   DISPH ๏ɺGSPH ๏ͱ͸
   • [Saitoh and Makino, 2013] Ͱ DISPH ๏͕ɺ[Inutsuka, 2002] Ͱ GSPH ๏͕։ൃ͞Εͨɻ
   DISPH ๏
   ઀৮ෆ࿈ଓ໘Ͱ࿈ଓͳѹྗΛΧʔωϧਪ
   ఆ͠ɺྲྀମͷඍ෼ํఔࣜΛղ͍͍ͯΔɻ
   ਓ޻೪ੑ͕ඞཁɻ
   GSPH ๏
   ཻࢠಉ࢜ͷ૬ޓ࡞༻ͷܭࢉͷࡍʹ
   Riemannsolver Λ༻͍ΔɻࠓճͷൃදͰ͸
   [Cha and Whitworth, 2003] ʹΑΔ Case3
   ͷ GSPH Λ࢖༻ɹਓ޻೪ੑඞཁͳ͠ɻ
   ౬ઙ ୓޺, ৿ ਖ਼෉ (ஜ೾େֶ) DISPH, Godunov SPH, Godunov DISPH, IA December 2, 2022 4 / 22
   

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5. ֤छεΩʔϜʹ͍ͭͯ Godunov DISPH ๏
   զʑ͕։ൃͨ͠ GDISPH ๏ʹ͍ͭͯ
   • DISPH ๏ͱಉ͘͡ɺີ౓ͷ୅ΘΓʹ઀৮ෆ࿈ଓ໘Ͱ࿈ଓͳѹྗΛΧʔωϧਪఆ͠ɺ
   GSPH ๏ͷಋग़ʹ฿ͬͯີ౓ʹཅʹґଘ͠ͳ͍ SPH ํఔࣜΛ࡞੒͍ͯ͘͠ɻ
   ρ(r) = ΣjmjW(|rj − r|, h(r)) (3)
   GDISPH ๏ಋग़ʹ༻͍Δࣜ
   q(r) = ΣjmjujW(|rj − r|, h(r)) =
   P(r)
   γ − 1
   (4)
   F(r) ∼
   ∫
   F(r′)W(|r − r′|, h(r′))d3r′ (5)
   ѹॖੑྲྀମͷӡಈํఔࣜɺΤωϧΪʔํఔࣜ
   dv(r)
   dt
   = −
   1
   ρ(r)
   ∇P(r) (6)
   du(r)
   dt
   = −
   P(r)
   ρ(r)
   ∇ · v(r) (7)
   ౬ઙ ୓޺, ৿ ਖ਼෉ (ஜ೾େֶ) DISPH, Godunov SPH, Godunov DISPH, IA December 2, 2022 5 / 22
   

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6. ֤छεΩʔϜʹ͍ͭͯ Godunov DISPH ๏
   GDISPH ๏ʹ͓͚Δ Case3 ܕͷӡಈํఔࣜɺΤωϧΪʔํఔࣜ
   ˙
   vi = −ΣjmjujP∗
   ij
   u∗
   ij
   [
   1
   q2
   i
   ∇iW(ri − rj, hi) +
   1
   q2
   j
   ∇iW(ri − rj, hj)
   ]
   ˙
   ui = −ΣjmjujP∗
   ij
   u∗
   ij
   [
   v∗
   ij
   − vi
   ]
   [
   1
   q2
   i
   ∇iW(ri − rj, hi) +
   1
   q2
   j
   ∇iW(ri − rj, hj)
   ] (8)
   ఴ͑ࣈʹ*ͷ෇͍ͨ෺ཧྔͷܭࢉํ๏ʹ͸ࣗ༝౓Λ࣋ͨͤΔɻ
   GSPH ๏Ͱ͸ɺriemann solver ͷղɻ
   ྫͱͯ͠ɺu∗
   ij
   = ui, v∗
   ij
   = vi+vj
   2
   ͱ͠ɺP∗
   ij
   ͸ riemann solver ͔ΒٻΊΔύλʔϯΛ
   GDISPH Case3 2 2 ͱ໊͚ͮΔɻ(ຊൃදͰ͸͜Ε͕ॏཁͳ GDISPH ͷεΩʔϜ)
   Χʔωϧͷۭؒඍ෼ʹ͸ɺ[Garc´
   ıa-Senz et al., 2012] ͕ߟҊͨ͠ IA Λ࢖༻͢Δ͔൱͔ͷࣗ༝
   ౓͕ଘࡏɻ
   ౬ઙ ୓޺, ৿ ਖ਼෉ (ஜ೾େֶ) DISPH, Godunov SPH, Godunov DISPH, IA December 2, 2022 6 / 22
   

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7. ֤छεΩʔϜʹ͍ͭͯ Integral Approach
   Integral Approach(IA) ͱ͸
   • [Garc´
   ıa-Senz et al., 2012] ͕೤֦ࢄ߲౳ʹݱΕΔ 2 ֊ඍ෼߲ͷܭࢉʹ࢖༻͞Ε͍ͯͨํ๏
   ([Brookshaw, 1985]) ΛҰ֊ඍ෼ʹԠ༻
   • 1 ֊ඍ෼ͰݱΕΔΧʔωϧͷۭؒඍ෼Λผͷ΋ͷʹม͑Δ͚ͩͰΑ͍ɻ
   ∇iW(ri − rj, hi) ∼ τ−1
   i
   (rj − ri)Wij(hi)
   ∇iW(ri − rj, hj) ∼ τ−1
   j
   (rj − ri)Wij(hj)
   τ(r) =
   [∫
   (r′ − r) ⊗ (r′ − r)W(|r − r′|, h(r))
   ]
   d3r′
   (9)
   • IA ʹΑΓ෺ཧྔͷ 1 ֊ඍ෼ͷਫ਼౓͕ྑ͘ͳΓɺۭؒ 0 ࣍ͷޡࠩ΋খ͘͞ͳΔ͜ͱ͕ใࠂ
   ͞Ε͍ͯΔɻ
   ౬ઙ ୓޺, ৿ ਖ਼෉ (ஜ೾େֶ) DISPH, Godunov SPH, Godunov DISPH, IA December 2, 2022 7 / 22
   

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8. ςετܭࢉ ࣮ݧͷ໨త
   ࣮ݧͷ໨త
   • SPH ๏ɺDISPH ๏
   ▶ Χʔωϧͷۭؒඍ෼Λ IA ʹ͢Δ͔൱͔
   • Case3 ͷ GSPH ๏
   ▶ Χʔωϧͷۭؒඍ෼Λ IA ʹ͢Δ͔൱͔
   ▶ riemann solver ͷߴਫ਼౓Խ͢Δ͔൱͔ɺ͢ΔͳΒྲྀ଎੍ݶؔ਺ΛԿʹ͢Δ͔
   • Case3 ͷ GDISPH ๏
   ▶ Χʔωϧͷۭؒඍ෼Λ IA ʹ͢Δ͔൱͔
   ▶ ఴ͑ࣈʹ*ͷ෇͍ͨ෺ཧྔΛ riemann solver Ͱղ͔͘൱͔ɺղ͔ͳ͍ͳΒԿʹ͢Δ͔
   ▶ riemann solver ͷߴਫ਼౓Խ͢Δ͔൱͔ɺ͢ΔͳΒྲྀ଎੍ݶؔ਺ΛԿʹ͢Δ͔
   riemann solver ͷߴਫ਼౓Խ͸ [Cha and Whitworth, 2003] ͷ΍ΓํͰߦͬͨɻ
   ֤εΩʔϜʹର༷ͯ͠ʑͳ૊Έ߹Θͤͷύλʔϯ͕ଘࡏɻ͜ͷதͰ࠷΋༏ΕͨεΩʔϜΛબ
   ఆ͢ΔͨΊɺ໿ 150 ݸͷύλʔϯʹؔͯ͠ riemann ໰୊,2D ੩ਫѹฏߧ,2D Kelvin-Helmholtz
   ෆ҆ఆੑͷܭࢉΛߦͬͨɻͦͷதͰ͍͔ͭ͘ͷεΩʔϜΛϐοΫΞοϓ͠ɺͦͷಛੑΛൺֱ
   ݕ౼͢Δɻ
   ౬ઙ ୓޺, ৿ ਖ਼෉ (ஜ೾େֶ) DISPH, Godunov SPH, Godunov DISPH, IA December 2, 2022 8 / 22
   

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9. ςετܭࢉ Riemann ໰୊
   Riemann ໰୊
   ڧ͍িܸ೾͕ൃੜ͢Δ໰୊
   {
   ρ = 1, P = 1000, v = 0 x < 0 ͷͱ͖
   ρ = 1, P = 0.1, v = 0 x > 0 ͷͱ͖
   (10)
   ີ౓ ѹྗ ଎౓
   ౬ઙ ୓޺, ৿ ਖ਼෉ (ஜ೾େֶ) DISPH, Godunov SPH, Godunov DISPH, IA December 2, 2022 9 / 22
   

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10. ςετܭࢉ Riemann ໰୊
    ౬ઙ ୓޺, ৿ ਖ਼෉ (ஜ೾େֶ) DISPH, Godunov SPH, Godunov DISPH, IA December 2, 2022 10 / 22
    

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11. ςετܭࢉ 2D ੩ਫѹฏߧ
    2D ੩ਫѹฏߧ
    ॳظ৚݅ͷ··มಈ͠ͳ͍ͷ͕෺ཧతͳղ ѹྗ͸શྖҬͰҰఆɹ
    • Ի଎ Cs = 1.02 ͰܭࢉྖҬΛԣ੾Δ࣌ؒ໿ 1.0 ͷ 8 ഒͰ͋Δ t = 8.0 ·Ͱܭࢉ
    ౬ઙ ୓޺, ৿ ਖ਼෉ (ஜ೾େֶ) DISPH, Godunov SPH, Godunov DISPH, IA December 2, 2022 11 / 22
    

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12. ςετܭࢉ 2D ੩ਫѹฏߧ
    ౬ઙ ୓޺, ৿ ਖ਼෉ (ஜ೾େֶ) DISPH, Godunov SPH, Godunov DISPH, IA December 2, 2022 12 / 22
    

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13. 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໿ 2 ഒͷ t = 2.0 ·Ͱܭࢉ
    GSPH,GDISPH ͱ SPH,DISPH ͷൺֱͷͨΊɺbalsara switch ͸͋͑ͯ֎ͯ͠ܭࢉ
    ॳظ৚݅ ༩͑ͨઁಈ
    ౬ઙ ୓޺, ৿ ਖ਼෉ (ஜ೾େֶ) DISPH, Godunov SPH, Godunov DISPH, IA December 2, 2022 13 / 22
    

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14. 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
    (11)
    ౬ઙ ୓޺, ৿ ਖ਼෉ (ஜ೾େֶ) DISPH, Godunov SPH, Godunov DISPH, IA December 2, 2022 14 / 22
    

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15. ςετܭࢉ 2D Kelvin-Helmholtz ෆ҆ఆੑ
    ౬ઙ ୓޺, ৿ ਖ਼෉ (ஜ೾େֶ) DISPH, Godunov SPH, Godunov DISPH, IA December 2, 2022 15 / 22
    

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16. ςετܭࢉ 2D Kelvin-Helmholtz ෆ҆ఆੑ
    • ࠨ͸ਓ޻೪ੑͷύϥϝʔλ α = 0.5ɺ
    ӈ͸ α = 3ɻӈͷ΄͏͕ਓ޻೪ੑ͕
    ڧ͍ɻ
    • ਓ޻೪ੑ͕ڧ͍ → γΞྖҬͰͷਓ޻
    ೪ੑ͕݁Ռతʹେ͖͘ͳΔ → ෆ҆ఆ
    ੑͷ੒௕͕཈੍
    ౬ઙ ୓޺, ৿ ਖ਼෉ (ஜ೾େֶ) DISPH, Godunov SPH, Godunov DISPH, IA December 2, 2022 16 / 22
    

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17. ςετܭࢉ ·ͱΊ
    ·ͱΊ
    • GDISPH Case3 2 2
    ▶ Riemann ໰୊ɺ੩ਫѹฏߧ໰୊ʹ͓͍ͯ DISPH ͱಉ༷ͳ݁Ռ,Kelvin-Helmholtz ෆ҆ఆੑʹ
    ؔͯ͠͸ GSPH ͱಉ༷ͳ݁ՌɻSPH ͷεΩʔϜͱͯ͠ɺѹॖੑྲྀମͷ໰୊Λ͖ͪΜͱղ͚
    ͍ͯΔɻ
    ▶ ߴਫ਼౓Խͨ͠৔߹ɺ೪ੑ͕খ͘͞ͳΔ͜ͱ͕ݪҼͰڧ͍িܸ೾ԼͰͷৼಈ͕ൃੜͯ͠͠
    ·͏ɻ
    ▶ ߴਫ਼౓ԽʹΑͬͯ Kelvin-Helmholtz ෆ҆ఆੑͷ੒௕͕ଅਐ͞ΕΔɻओͳཁҼ͸೪ੑ͕খ͞
    ͘ͳͬͨ͜ͱͰγΞྖҬͷ೪ੑ͕খ͘͞ͳ͔ͬͨΒ
    • IA
    ▶ Kelvin-Helmholtz ෆ҆ఆੑͰ͸ɺIA ԽʹΑΓ੒௕͕ଟগଅਐ͞ΕΔɻ͔͠͠ɺ੩ਫѹฏߧ໰
    ୊͔Β DISPH ܥͷεΩʔϜͰ͸ IA ԽʹΑΓϊΠζΛर͍΍͘͢ͳ͍ͬͯΔɻ
    • SSPH,DISPH
    ▶ Kelvin-Helmholtz ෆ҆ఆੑʹ͓͍ͯɺಉ༷ͳ݁ՌɻGSPH,GDISPH ΑΓ΋ෆ҆ఆੑͷ੒௕͕
    ଅਐ͞Ε͍ͯΔ͕ɺ͔͔͍ͬͯΔ೪ੑ͕ GSPH,GDISPH ΑΓ΋খ͍͔͞Βɻ
    ౬ઙ ୓޺, ৿ ਖ਼෉ (ஜ೾େֶ) DISPH, Godunov SPH, Godunov DISPH, IA December 2, 2022 17 / 22
    

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18. sheer switch ͷվྑʹ޲͚ͯ ໰୊఺
    ౬ઙ ୓޺, ৿ ਖ਼෉ (ஜ೾େֶ) DISPH, Godunov SPH, Godunov DISPH, IA December 2, 2022 18 / 22
    

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19. sheer switch ͷվྑʹ޲͚ͯ ݪҼͷ༧૝
    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 ୓޺, ৿ ਖ਼෉ (ஜ೾େֶ) DISPH, Godunov SPH, Godunov DISPH, IA December 2, 2022 19 / 22
    

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20. sheer switch ͷվྑʹ޲͚ͯ վળࡦ
    վળࡦ
    • ෺ཧతͳઁಈʹΑΔෆ҆ఆੑͷ੒௕͸ڐ͠ɺ਺஋తͳϊΠζʹΑΔઁಈͷ੒௕͸཈͑Δɻ
    → balsara switch ͷڧ͞Λௐઅ͢Δ͜ͱͰ࣮ݱΛ໨ࢦ͢ɻ
    Balsara switch
    • ਓ޻೪ੑ߲ʹ 0.5(Fi + Fj) Λ͔͚Δɻ
    Fi =
    |∇i · vi|
    |∇i · vi| + |∇i × vi| + 0.0001ci/hi
    (12)
    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) (13)
    ౬ઙ ୓޺, ৿ ਖ਼෉ (ஜ೾େֶ) DISPH, Godunov SPH, Godunov DISPH, IA December 2, 2022 20 / 22
    

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21. sheer switch ͷվྑʹ޲͚ͯ վળࡦ
    DISPH
    ౬ઙ ୓޺, ৿ ਖ਼෉ (ஜ೾େֶ) DISPH, Godunov SPH, Godunov DISPH, IA December 2, 2022 21 / 22
    

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22. ࠓޙͷํ਑
    ࠓޙͷํ਑
    • ೪ੑྲྀମͷ Kelvin-Helmholtz ෆ҆ఆੑͷ෼ࢄؔ܎͔ΒɺO(h) ఔ౓ͷ೾௕ͷઁಈͷ੒௕Λ
    ແࢹͰ͖ΔΑ͏ͳύϥϝʔλ β ͷઃఆํ๏Λݟ͚ͭΔɻ
    • GSPH,GDISPH ΁ͷ Balsara switch ͷద༻Λߦ͏ɻ
    • ఆྔతͳධՁΛ༻͍Δ
    • ͕Μ͹Δ
    ౬ઙ ୓޺, ৿ ਖ਼෉ (ஜ೾େֶ) DISPH, Godunov SPH, Godunov DISPH, IA December 2, 2022 22 / 22
    

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23. ࠓޙͷํ਑
    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 ୓޺, ৿ ਖ਼෉ (ஜ೾େֶ) DISPH, Godunov SPH, Godunov DISPH, IA December 2, 2022 22 / 22
    

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24. ࠓޙͷํ਑
    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 ୓޺, ৿ ਖ਼෉ (ஜ೾େֶ) DISPH, Godunov SPH, Godunov DISPH, IA December 2, 2022 22 / 22
    

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25. ࠓޙͷํ਑
    Reference II
    [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.
    ౬ઙ ୓޺, ৿ ਖ਼෉ (ஜ೾େֶ) DISPH, Godunov SPH, Godunov DISPH, IA December 2, 2022 22 / 22
    

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26. ࠓޙͷํ਑
    ಋग़ͷࡍʹ༻͍ΔԾఆɾۙࣅ
    • ҎԼͷࣜΛຬͨ͢ Fij ͕ଘࡏ͢Δ
    ∫
    F(r)
    q2(r)
    W(|r − ri|, h(r))W(|r − rj|, h(r))d3r
    = F∗
    ij
    ∫
    1
    q2(r)
    W(|r − ri|, h(r))W(|r − rj|, h(r))d3r
    (14)
    • Case3 ͷܗʹ͢ΔͨΊʹҎԼͷۙࣅ͕੒Γཱͭͱ͢Δ
    [∇i − ∇j]W(|ri − r|, h(r))W(|rj − r|, h(r))
    ∼ ∇iW(|ri − r|, h(r))δ(|r − rj|) − δ(|r − ri|)∇jW(|rj − r|, h(r))
    (15)
    • Godunov SPH ๏Ͱ͸ɺࣜ (14) ʹ͓͍ͯ q(r) ͕ ρ(r) ʹஔ͖׵Θ͍ͬͯΔɻFij ͸ཻࢠ i
    ͱཻࢠ j ͷத৺෇ۙͰͷɺཻࢠ i ͱཻࢠ j ͷ෺ཧྔΛ༻͍ͨ Riemann solver ͷղ
    ౬ઙ ୓޺, ৿ ਖ਼෉ (ஜ೾େֶ) DISPH, Godunov SPH, Godunov DISPH, IA December 2, 2022 22 / 22
    

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27. ࠓޙͷํ਑
    GDISPH ΁ͷ Balsara switch ͷద༻
    • GSPHɺGDISPH ʹ Balsara switch ͷద༻Λߦͬͨྫ͸ͳ͍ɻ
    • DISPH ͸ (ඇਓ޻೪ੑ߲)+(ਓ޻೪ੑ߲) ʹཅʹ෼͔Ε͍ͯΔ͕ɺGDISPH ͸ 1 ͭͷ߲ʹ
    ·ͱ·͍ͬͯͯ෼཭͍ͯ͠ͳ͍ɻ
    զʑͷఏҊख๏
    • (DISPH ͷඇਓ޻೪ੑ߲)+0.5(F′
    i + F′
    j)(GDISPH ͷ߲ - DISPH ͷඇਓ޻೪ੑ߲) Λɺ
    GDISPH ͷࣜͱͯ͠༻͍Δɻ
    • F′
    i
    = F′
    j
    = 1(Balsara switch Φϑ) Ͱͨͩͷ GDISPH
    ౬ઙ ୓޺, ৿ ਖ਼෉ (ஜ೾େֶ) DISPH, Godunov SPH, Godunov DISPH, IA December 2, 2022 22 / 22
    

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28. ࠓޙͷํ਑
    GDISPH
    ౬ઙ ୓޺, ৿ ਖ਼෉ (ஜ೾େֶ) DISPH, Godunov SPH, Godunov DISPH, IA December 2, 2022 22 / 22
    

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