senior_thesis_slides

Transcript

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
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
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
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
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
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
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
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
9. ֤छεΩʔϜʹ͍ͭͯ Standard SPH ๏ SSPH িܸ೾؅໰୊ ڧ͍িܸ೾ (ॳظ৚݅Ͱ 104 ͷѹྗࠩ) ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 9 / 45
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
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
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
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
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
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
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
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
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
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
20. ֤छεΩʔϜʹ͍ͭͯ Density-Independent SPH ๏ DISPH িܸ೾؅໰୊ ڧ͍িܸ೾ (ॳظ৚݅Ͱ 104 ͷѹྗࠩ) ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 20 / 45
21. ֤छεΩʔϜʹ͍ͭͯ Density-Independent SPH ๏ • ද໘ுྗͷޮՌ͕ͳ͘ͳ͍ͬͯΔ ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 21 / 45
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
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
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
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
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
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
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
29. ֤छεΩʔϜʹ͍ͭͯ Godunov SPH ๏ ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 29 / 45
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
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
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
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
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
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
36. ςετܭࢉ Riemann ໰୊ Godunov DISPH িܸ೾؅໰୊ ڧ͍িܸ೾ (ॳظ৚݅Ͱ 104 ͷѹྗࠩ) ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 36 / 45
37. ςετܭࢉ 2D ѹྗฏߧ • ද໘ுྗʹΑΔޮՌ͸ݟΒΕͳ͍ • ෺ཧతͳղͱͳ͍ͬͯΔ ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 37 / 45
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
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
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
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
42. ςετܭࢉ 2D Kelvin-Helmholtz ෆ҆ఆੑ ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 42 / 45
43. ςετܭࢉ 2D Kelvin-Helmholtz ෆ҆ఆੑ ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 43 / 45
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
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
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
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
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
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
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
51. ݁࿦ িܸ೾؅໰୊ ڧ͍িܸ೾ (ॳظ৚݅Ͱ 104 ͷѹྗࠩ) ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 45 / 45
52. ݁࿦ ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 45 / 45
53. ݁࿦ ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 45 / 45
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
55. ݁࿦ ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 45 / 45
56. ݁࿦ ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 45 / 45
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
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
59. ݁࿦ ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 45 / 45
60. ݁࿦ ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 45 / 45
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
62. ݁࿦ ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 45 / 45
63. ݁࿦ ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 45 / 45