Discontinuous Galerkin Based High Order Methods for Semiclassical Boltzmann-Bgk Equation with Applications to Quantum Thermal-Fluid Transports (I-Iii)

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Project title

Code/計畫編號

NSC102-2221-E019-067-MY3

Translated Name/計畫中文名

半經典波茲曼模式方程之不連續賈樂金高階方法及其在量子熱流傳輸之應用(I-III)

Project Coordinator/計畫主持人

Juan-Chen Huang

Funding Organization/主管機關

National Science and Technology Council

Co-Investigator(s)/共同執行人

楊照彥

Department/Unit

Department of Merchant Marine

Website

https://www.grb.gov.tw/search/planDetail?id=8111840

Year

2014

Start date/計畫起

01-08-2014

Expected Completion/計畫迄

31-07-2015

Bugetid/研究經費

745千元

ResearchField/研究領域

機械工程

"於此三年計晝吾人擬研發新世代基於不連續賈樂金(discontinuous Galerkin)之高階方法來解半經典波茲曼模式方程。主要發展發法為Runge-Kutta型不連續賈樂金方法和強型重建修正高階方法（correction procedure via reconstruction， CPR)，然後將發展出的新的高階方法應用至半經典奈米尺度熱傳現象，如半導體内電子傳輸及奈米結構熱電材料之聲子熱傳。加添型Runge-Kutta時間積分算則和半經典橢圓型模式也將一併考量。吾人將每年的工作項目和目標分年描述如下。吾人研究團隊在過去已經成功地完成了不震盪有限差分高階震波捕捉法，應用於氣體動力學、空氣動力學、稀薄氣體動力學、相對論氣體動力學及量子熱流傳輸。這些已含了 TVD、ENO和WENO方法，屬 1980〜2000年之方法，使用了較廣較多的算則格點，不利於邊界條件。本計晝發展的是較局部使用元素或格胞内之高階力矩量以達成高精確度。相信在這三年計晝執行完後，吾人將擁有解半經典波茲曼模式方程之新世代高階方法，並在國際學術界之奈米尺度熱流傳輸和半經典量子流體及稀薄量子氣動力問題具有競爭力。成果除了發表多篇論文於國際著名期刊，並將應用到高科技的奈米熱流傳輸問題上。第一年（2013/08-2014/07) 吾人建構一組總變量穩定、不連續賈樂金(discontinuous Galerkin，DG)方法以解半經典波茲曼模式（semiclassical Boltzmann-BGK)方程。探討方法和理論推導及建構一維和多維算則。首先應用速度分立座標法以離散動量（或速度）空間而導致一組在物理空間之聯立具源項之雙曲線守怪偏微分方程。然後應用弱型（weak form)不連續賈樂金方法以離散空間而達到一時間連續之半離散方程組，最後經由多階段的Runge-Kutta時間積分程式以達成時間邁進。為了防止高強度震波時產生震盪及不穩定，吾人將使用TVB 和WENO限制子來標示搗亂胞元（troubled cell)而加以修正。以適當的範例來測試及顯示所發展的高階方法。最後將一維方法擴展至二維方法。此方法計算量很大，但可適用之熱流參數(Reynolds數，Mach數，Knudsen數）範圍較廣泛。可以含括連續體流、滑動流、過渡流和自由流等流域的計算。主要的困難在於數學公式的複雜度增加，從Maxwell-Boltzmann統計（一般原子-分子）至Bose-Einstein統計（聲子）及 Fermi-Dirac統計（電子）。在奈米尺度下，傅立葉熱傳定律不能適用，吾人將應用至奈米尺度熱流傳輸問題上。吾人目標完成二至三篇主要期刊論文。第二年（2014/08-2015/07) 吾人將建構發展和不連續賈樂金方法很相關的重建修正法（correction procedure via reconstruction， CPR)之高階方法以解半經典波茲曼模式（semiclassical Boltzmann-BGK)方程。首先應用速度分立座標法以離散動量（或速度）空間而導致一組在物理空間連續之具源項雙曲型偏微分方程組。其次應用重建修正法 (CPR)以離散物理空間而達到一組時間連續之半離散常微分方程組，最後使用TVB Runge-Kutta多階段時間積分而達成整個高階數值方法的建構。吾人將測試CPR 方法和DG法及譜差分（spectral difference)法之間的關聯性也一併探討。CPR 的優點乃在於解守怪律於微分式（strong form)而非積分式（weak form)，故較簡單。一維及二維的方法並應用至奈米尺度熱流傳輸用。在應用方面除了半經典氣動力學問題(如二維翼剖面流場），如在奈米碳管内之電子/聲子傳輸。本年度目標完成二至二篇主要期刊論文。第三年（2015/08-2016/07) 以前兩年所建立的基礎，主要是一維和二維的不連續賈勒金法和CPR方法，第三年將其擴展至三維方法而達到一實用方法及應用至廣泛的奈米尺度熱流傳輸問題的模擬預測工具。同樣地結合分立座標法，不連續賈勒金法和CPR方法及多階段Runge-Kutta 時間積分法，而達成半經典波茲曼模式方程式之三維高階解法算則。有此工具，吾人將可解實際複雜三維問題。最後將整理三年内完成的工作，對三種統計粒子之傳輸其物理機制差異作系統性分析。本年度目標完成二至三篇主要期刊論文。在應用方面除了半經典氣動力學問題(如三維翼流場），如在奈米碳管内之電子/聲子傳輸。成果發表二至三篇國際著名期刊論文並應用到奈米科技傳輸問題上。""Summary In this three-year proposed project, we intend to first develop and implement the new generation high order methods consisted of the total variation stable Runge-Kutta discontinuous Galerkin (DG) methods and high order correction procedure via reconstruction method for solving the semiclassical Boltzmann-BGK equation in phase space. Then we apply the resulting high numerical methods to modern nanoscale thermal-fluid transports, e.g., electron transport in semiconductors and phonon energy transfer in nanostructured thermoelectric materials. Other related topics will be covered including semiclassical ellipsoidal model and additive Runge-Kutta methods. We describe below each year’s main themes and objectives, separately. In the past we have successfully developed finite difference high resolution shock capturing schemes and applied to computational aerodynamics, rarefied gas dynamics, ideal quantum gas dynamics, relativistic gas dynamics and quantum hydrodynamics (see references below). All these are based on traditional methods which usually achieve high-order accuracy by using a wide stencil, hence they may lose accuracy in a fairly large region near shocks, and they usually are difficult to apply in complicated geometries and/or boundary conditions. The new generation high order methods (after year 2000), including discontinuous Galerkin, spectral difference, spectral volume and correction procedure via reconstruction (CPR) methods which are more local, in the sense that higher orders are achieved by more moments in a cell rather than using neighboring cells. We believe with the completion of this three-year project and with the new generation high order methods for solving the semiclassical Boltzmann-BGK equation in multiple dimensions at hand, we shall establish strong competitive capability internationally in the fields of nanoscale thermal-fluid transport and semiclassical hydrodynamics/rarefied gas dynamics. Several major papers will result from this project each year. First Year (08/2013-07/2014) Abstract - We construct a class of TVB (total variation bounded) discontinuous Galerkin finite element methods for solving semiclassical Boltzmann-BGK equation. We first apply the discrete ordinate method to discretize the momentum (or velocity) space to yield a set of hyperbolic conservation law with still source term. Then, the TVD Runge-Kutta discontinuous Galerkin method is applied to produce a new class of high order methods for solving the semiclassical Boltzmann-BGK equation. Computational examples to test and to illustrate the methods are provided. Extension to two spatial dimensions will be also included. The newly developed additive Runge-Kutta method will be also considered. Applications to semiclassical thermal-fluid transport problems where the classical Navier-Stokes viscous law or Fourier heat conduction law is inadequate will be simulated. Two to three journal papers will be completed. Second Year (08/2014-07/2015) Abstract - We construct and implement a class of high order methods by correction procedures using reconstruction (CPR) for solving the semiclassical Boltzmann-BGK equation which unify several existing methods such as spectral volume and spectral difference and it recovers the discontinuous Galerkin methods. We first apply the discrete ordinate method to discretize the momentum (or velocity) space to yield a set of hyperbolic conservation law with still source term. Then, the CPR method is applied to produce a new class of high order methods for solving the semiclassical Boltzmann-BGK equation. Computational examples to test and to illustrate the methods are provided. Extension to two spatial dimensions (unstructured mesh or element) will be also included. Application to semiclassical transport problems where the classical Navier-Stokes viscous law or Fourier heat conduction law is inadequate will be simulated. Two to three journal papers will be completed. Third Year (08/2015-07/2016) Abstract - Based on the foundation and results of previous two years, we consider three-dimensional construction and implementation of high order methods by discontinuous Galerkin (DG) and correction procedures via reconstruction (CPR) for solving the semiclassical Boltzmann-BGK equation. We first apply the discrete ordinate method to discretize the momentum (or velocity) space to yield a set of hyperbolic conservation law with still source term. Then, the DG and/or CPR method is applied to produce a new class of high order methods for solving the semiclassical Boltzmann-BGK equation. Computational examples to test and to illustrate the 3-D methods will be provided. Applications to practical and engineering semiclassical thermal-fluid transport problems will be simulated. Two to three journal papers will be completed."

Keyword(s)

半經典波茲曼傳輸
半經典量子熱流
不連續賈樂金方法
重建修正高階方法
奈米尺度載子傳輸
稀薄量子氣動力學
Semiclassical Boltzmann-BGK equation
Nanoscale thermal-fluid transport
Rarefied quantum gas flows
Discontinuous Galerkin methods
Correction procedure via reconstruction
TVD Runge-Kutta
Additive Runge-Kutta methods

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Discontinuous Galerkin Based High Order Methods for Semiclassical Boltzmann-Bgk Equation with Applications to Quantum Thermal-Fluid Transports (I-Iii)

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