1. Gabatarwa
Maganin ƙididdiga na ma'auni na Boltzmann yana gabatar da ƙalubale masu mahimmanci saboda girman girmansa mai girma (7D don aikace-aikacen 3D), yankin gudu mara iyaka, da kuma mai aikin karo mara layi mai cike da ƙididdiga wanda ke buƙatar kimanta haɗin kai mai girma biyar. Babban abin da ake buƙata shine kiyaye yawa, motsi, da kuzari yayin karo. Wannan takarda ta ginu akan hanyar ƙididdiga ta ƙayyadaddun ƙididdiga da Gamba da Tharkabhushanam suka haɓaka, yana tsawaita shi zuwa daidaiton mataki na biyu kuma yana inganta shi don yanayin lissafi mai girma (HPC). Hanyar tana amfani da tsarin mai aikin karo da aka canza ta Fourier, yana sake tsara shi azaman haɗin kai mai nauyi, kuma yana tilasta kiyayewa ta hanyar matsalar ingantawa mai ƙuntatawa.
2. Hanyar Aiki
2.1. Tsarin Hanyar Spectral
Babban ƙirƙira yana cikin aiki akan siffar rauni na ma'auni na Boltzmann da amfani da canje-canjen Fourier. Haɗin kai $Q(f,f)$ an canza shi zuwa haɗin kai mai nauyi a cikin sararin Fourier: $\hat{Q}(\xi) = \int_{\mathbb{R}^d} \hat{f}(\xi_+) \hat{f}(\xi_-) \mathcal{B}(\xi, \xi_*) d\xi_*$, inda $\xi$ shine madaidaicin Fourier, kuma $\mathcal{B}$ shine ƙirar da aka samo daga sashin giciye na karo. Wannan hanyar tana guje wa kimanta kai tsaye na haɗin kai mai girma a cikin sararin zahiri.
2.2. Tilasta Kiyayewa Ta Hanyar Ingantawa
Ƙididdiga na Spectral na iya karkata daga kiyaye masu canjin karo (yawa $\rho$, motsi $\rho u$, kuzari $\rho E$). Hanyar tana tilasta kiyayewa ta hanyar warware matsalar ingantawa mai ƙuntatawa bayan karo: nemo rarraba $\tilde{f}$ mafi kusa da fitarwar Spectral $f^*$ a ma'anar $L^2$, bisa ga $\int \phi(\mathbf{v}) \tilde{f} d\mathbf{v} = \int \phi(\mathbf{v}) f_0 d\mathbf{v}$, inda $\phi(\mathbf{v}) = \{1, \mathbf{v}, |\mathbf{v}|^2\}$. Wannan yana tabbatar da cewa filayen macroscopic suna haɓaka daidai.
2.3. Tsawaita Mataki na Biyu a Sarari da Lokaci
An tsawaita hanyar asali don cimma daidaiton mataki na biyu a cikin sarari da lokaci, yana ɗaukar grid ɗin da ba su da daidaito. Wannan yana ƙunshe da ƙayyadaddun sarari mafi girma (misali, tsare-tsare na ƙididdiga/tsarin bambanci) da tsare-tsaren haɗin kai na lokaci kamar hanyoyin Runge-Kutta, yana inganta amincin mafita sosai don magudanan da suka ƙunshi rikitarwa.
3. Aiwatar da Lissafi Mai Girma
3.1. Rarrabawar Ƙwaƙwalwar Ajiya da Wuri
Babban fa'ida ga HPC shine wurin kalmar karo. Kimanta mai aikin karo a wani batu a cikin sararin zahiri ya dogara ne kawai akan rarraba gudu a wannan batu, ba akan maki na sararin da ke kusa ba. Wannan yana ba da damar dabarar rarraba yanki kai tsaye: za a iya raba sararin zahiri a cikin nodes/cores na kwamfuta tare da ƙaramin ƙarar saƙo, saboda kawai bayanan iyaka don matakin jujjuyawar ne kawai ake buƙatar musayar.
3.2. Gwaje-gwajen Ma'auni akan Babban Kwamfuta na Lonestar
An yi gwaje-gwajen ma'auni na farko akan babban kwamfuta na Lonestar a Cibiyar Lissafi ta Texas Advanced Computing Center (TACC). Takardar tana nuna cewa waɗannan gwaje-gwajen sun nuna ingancin rarrabawar ƙwaƙwalwar ajiya da iyawar algorithm ɗin, ko da yake ba a cika bayyana takamaiman ma'auni na ingancin layi daya (ma'auni mai ƙarfi/rauni) ba a cikin abin da aka samo.
4. Cikakkun Bayanai na Fasaha da Tsarin Lissafi
Ma'auni na Boltzmann shine: $\frac{\partial f}{\partial t} + \mathbf{v} \cdot \nabla_{\mathbf{x}} f = Q(f,f)$. Tushen hanyar Spectral shine siffar canjin Fourier don nau'in Maxwell da ƙarfi mai wuyar canzawa. Mai aikin karo a cikin sararin Fourier ya zama haɗin kai, amma tare da nauyin $\mathcal{B}$ wanda gabaɗaya yana hana amfani da Canjin Fourier Mai Sauri (FFT) don cimma rikitarwa na $O(N^d \log N)$, yana haifar da ayyuka $O(N^{2d})$. Hanyar tana amfani da kayan aikin FFT a cikin yankin lissafi tare da mai tsawaitawa don tabbatar da haɗuwa zuwa mafita mai ci gaba, bin tsarin a cikin sararin Sobolev.
5. Sakamako da Aikace-aikace
5.1. Matsalar Gargadi da Layer na Iyaka Ya Haifar
An yi amfani da ƙarfin lissafi mai ƙarfi na wannan hanyar don bincika matsalar gargadi da Layer na Iyaka ya haifar wanda ba za a iya bayyana shi da lissafin ruwa na gargajiya ba (ma'auni na Navier-Stokes). Wannan shine yanayin motsin iskar gas da ba a taɓa gani ba inda lambar Knudsen ba ta da mahimmanci. Hanyar ƙididdiga ta ƙayyadaddun ƙididdiga, ba tare da hayaniyar ƙididdiga ba, ta dace musamman don ɗaukar tasirin rashin daidaito da cikakken tsarin irin waɗannan abubuwan mamaki, waɗanda ke da mahimmanci a cikin motsin iska mai tsayi da magudanan ƙananan sikelin.
6. Tsarin Bincike: Nazarin Shari'ar da Ba ta Ƙunshi Code ba
Shari'a: Tabbatar da Kaddarorin Kiyayewa a cikin Gwajin Sakewa zuwa Daidaito. 1. Saitin Matsala: Fara yanki na sarari 1D tare da rarraba gudu mara daidaito (misali, haɗa Maxwellians biyu a yanayin zafi daban-daban). Yi amfani da sharuɗɗan iyaka na lokaci-lokaci don ware tsarin karo. 2. Kwaikwayo: Gudanar da mai warwarewa na Boltzmann na Spectral tare da kashe matakin tilasta kiyayewa. Lura da haɓakar jimillar yawa, motsi, da kuzari. Lura da karkatarwa. 3. Shiga Tsakani: Kunna matakin ingantawa mai ƙuntatawa. Sake gudanar da kwaikwayon. 4. Bincike: Kwatanta gudun biyu. Babban alamar aiki shine matakin kiyayewa na injin ($\sim 10^{-14}$) na masu canzawa a cikin gudu na biyu, sabanin karkatarwa da ake iya auna a farkon. Wannan yana tabbatar da ainihin tsarin kiyayewa, babban fa'ida akan wasu hanyoyin Monte Carlo inda kiyayewa ke cika kawai ta hanyar ƙididdiga.
7. Aikace-aikace na Gaba da Hanyoyi
- Magudanan Komawa Hypersonic: Yin samfuri na garkuwar zafi na jirgin sama inda manyan abubuwan mamaki da rashin daidaiton sinadarai na zafi suka yi yawa.
- Tsarin Injiniya na Lantarki na Micro (MEMS): Yin kwaikwayon magudanan iskar gas a cikin ƙananan na'urori inda tasirin raguwa ya fi rinjaye.
- Kimiyyar Plasma: Tsawaita tsarin zuwa ma'auni na Boltzmann don barbashi masu caji, masu dacewa a cikin haɗakarwa da tuƙi na sararin samaniya.
- Haɗin kai na Algorithm-Hardware: Bincika aiwatarwa akan GPUs da masu haɓaka AI don amfani da tsarin haɗin kai na kamanceceniya da tsarin haɗin kai.
- Hanyoyin Haɗin kai: Haɗa wannan mai warwarewa mai ƙayyadaddun ƙididdiga a yankuna masu girma tare da masu warwarewa na ruwa masu sauri a yankunan daidaito don matsalolin sikelin da yawa.
8. Nassoshi
- Gamba, I.M., & Tharkabhushanam, S. (2009). Hanyoyin Spectral-Lagrangian don samfuran karo na jihohin ƙididdiga marasa daidaito. Jaridar Lissafin Lissafi.
- Bobylev, A.V. (1976). Hanyar canjin Fourier don ma'auni na Boltzmann. Lissafin Lissafi da Lissafin Lissafi na USSR.
- Pareschi, L., & Perthame, B. (1996). Hanyar Spectral ta Fourier don ma'auni na Boltzmann masu kama da juna. Ka'idar Sufuri da Lissafin Ƙididdiga.
- Pareschi, L., & Russo, G. (2000). Maganin ƙididdiga na ma'auni na Boltzmann I: Ƙididdiga mai daidaiton ƙididdiga na mai aikin karo. Jaridar SIAM akan Nazarin Ƙididdiga.
- Ibragimov, I., & Rjasanow, S. (2002). Maganin ƙididdiga na ma'auni na Boltzmann akan grid ɗin daidaito. Lissafi.
- Bird, G.A. (1994). Motsin Gas na Kwayoyin Halitta da Kwaikwayon Kai Tsaye na Magudanan Iskar Gas. Clarendon Press. (Don kwatanta DSMC).
- Cibiyar Lissafi ta Texas Advanced Computing Center (TACC). (2023). Babban Kwamfuta na Lonestar. https://www.tacc.utexas.edu/systems/lonestar
9. Binciken Kwararru & Bita Mai Mahimmanci
Babban Fahimta: Wannan aikin ba wani ƙarin ci gaba ne kawai ga mai warwarewa na Boltzmann ba; yana da ƙirar injiniya na hanyar ƙididdiga mai kyau ta lissafi don zamanin lissafi na exascale. Marubutan sun gano kuma sun yi amfani da wurin sarari na mai aikin karo na Spectral—wani abu da ake yawan watsi da shi—a matsayin mabuɗin ingantaccen layi daya mai yawa. Wannan yana mai da dabbobi masu ban tsoro na lissafi na $O(N^{2d})$ zuwa matsalar da ta dace da rarrabawar yanki mai kyau, yana magance kai tsaye la'anar "girman girma" da suka ambata.
Kwararar Ma'ana: Ma'ana tana da ban sha'awa: 1) Fara da ainihin ƙididdiga mai girma, mai tsare-tsare (Gamba & Tharkabhushanam). 2) Gano toshewar sa (farashin lissafi) da ƙarfinsa na ɓoye (wurin sarari). 3) Ƙirƙirar tsawaita mataki na biyu don amincin aiki. 4) Sake tsara aiwatarwa a kusa da ƙarfi don HPC, ta amfani da wurin don rage saƙo, babban mai kashe iyawa. 5) Tabbatarwa ta hanyar magance matsalar da ke nuna ƙimar ƙima ta musamman ta hanyar: wani abin mamaki mara daidaito wanda ba a iya gani ga CFD na gargajiya. Wannan misali ne na littafi don binciken lissafi mai motsa matsalar.
Ƙarfi & Kurakurai: Ƙarfi: Haɗin kai na kiyayewa mai tsauri (ta hanyar ingantawa) tare da ƙirar HPC yana da ƙarfi. Yana ba da madadin ƙayyadaddun ƙididdiga, ƙaramin hayaniya ga DSMC don matsalolin lokaci da ƙananan Mach, yana cike da mahimmin rami. Aikace-aikacen ga abin mamaki na Layer na Iyaka zaɓi ne na tabbatar da ra'ayi wanda ke nuna dacewa ga hypersonics da MEMS. Kurakurai: Giwa a cikin ɗaki har yanzu ita ce ma'auni na $O(N^{2d})$ a cikin sararin gudu. Yayin da layi daya na sarari ya warware, "bangon sararin gudu" don kwaikwayon 3D mai girma har yanzu yana da ƙarfi. Takardar tana nuna amma ba ta cika magance wannan ba. Bugu da ƙari, matakin ingantawa mai ƙuntatawa, yayin da yake da kyau, yana ƙara ƙarin nauyin lissafi mara mahimmanci a kowane lokaci wanda ba a ƙididdige shi ba daidai da lissafin karo da kansa. Ta yaya wannan yake ma'auni?
Fahimta Mai Aiki: 1. Ga Masu Aiki: Wannan hanyar ya kamata ta kasance a cikin ɗan gajeren jerin ku don yin kwaikwayon ƙananan-zuwa-matsakaicin lambar Knudsen magudanan inda cikakkun bayanai da kiyayewa suke da mahimmanci, kuma kuna da damar samun albarkatun HPC masu yawa. Ba maye gurbin gabaɗaya ba ne don DSMC ko masu warwarewa na NSF, amma kayan aiki masu daidaito don takamaiman matsaloli masu wahala. 2. Ga Masu Bincike: Makomar ta ta'allaka ne a kai hari ga rikitarwa na $O(N^{2d})$. Bi jagorar ayyuka kamar waɗanda ke kan mai aikin Fokker-Planck-Landau da aka ambata a cikin takardar. Bincika hanyoyin multipole masu sauri, matrices masu matsayi, ko maye gurbin koyo mai zurfi (wanda aka yi wahayi ta hanyar nasarar samfura kamar Masu Aikin Fourier Neural) don kimanin haɗin kai mai nauyi. Ci gaba na gaba zai kasance a cikin karya wannan shingen rikitarwa yayin riƙe kiyayewa. 3. Ga Cibiyoyin HPC: Wurin da aka nuna yana sa wannan algorithm ya zama ɗan takara mai kyau ga tsarin gine-ginen GPU da na bambancin. Zuba jari a cikin jigilar sa da ingantawa na iya haifar da aikace-aikacen tuta don lissafin kimiyyar lissafi.
A ƙarshe, Haack da Gamba sun ba da ci gaban injiniya mai mahimmanci ga masu warwarewa na Boltzmann mai ƙayyadaddun ƙididdiga. Sun yi nasarar canza wani algorithm mai sarƙaƙƙiya daga fagen "lissafi mai ban sha'awa" zuwa "kayan aikin HPC na aiki". An ba da sandar yanzu ga al'umma don magance ainihin rikitarwar algorithm ɗin da ya rage, mai yuwuwa ta hanyar haɗin kai tare da sabbin ci gaban lissafi da koyon injin.