頻散關係得以保持之緊緻方法─求解光電及熱流方程式

On the dispersion-relation-preserving compact schemes for predicting electromagnetic wave, incompressible heat and fluid flow equations

邱柏雄(Pao-Hsiung Chiu)

指導教授 ： 許文翰

國立臺灣大學/工學院/工程科學及海洋工程學研究所/博士(2009年)

https://doi.org/10.6342/NTU.2009.00562

摘要

本論文提出能保有較佳頻散關係，且能增進對流項之穩定性及頻散準確性之對流項離散格式。論文中所提出之離散格式均經由嚴謹的頻散及消散分析。為了驗證論文中所提出的格式，本論文測試了許多具實解及典型的測試問題。由結果可知，本論文所提出之格式，在所有的測試問題中均能保有相當好的收斂斜率。
本論文接著將所提出之頻散關係得以保持之格式應用於求解非線性淺水波、光電電磁波以及不可壓縮流體。對於非線性淺水波，本論文量身訂做出一個能有效求解的疊代數值格式，該方法是為將原始之非線性淺水波方程式轉型為一對流─反應方程及赫姆霍茲方程。為了得到頻散準確的解，本論文使用了六階準確的頻散關係得以保持之格式來離散對流項，並搭配了六階的緊緻格式來求解赫姆霍茲方程。最後，為了保持非線性淺水波的漢米爾頓結構，本論文特地使用了二階及四階準確的辛時間積分算子。為了驗證此方法的可行性，本論文測試了包含行進波的測試問題。最後，本論文比較了目前的方法及完全可積粒子法、局部不連續Galerkin方法之準確度、需要的計算時間以及時間和空間的收斂斜率。
對於光電電磁波，本論文提出了一能求解二維及三維之非交錯格點求解器。為了能夠解決奇、偶震盪的問題並節省計算時間，本論文提出了一顯式的緊緻格式來離散一次微分項。為了保持其漢米爾頓結構，本論文特地使用了辛時間積分算子。藉由求解二維及三維TM模態的光電電磁波方程，證實本論文所提出之求解器的準確性及可行性。
對於不可壓縮流體，本論文提出一能在非錯離格點上有效求解的無散度補償方法。該方法的核心思想是在動量方程式中加入一經嚴謹推導出之補償無散度的源項，進而滿足無散度的拘束條件。使用本論文所提出之方法，可以避免直接求解無散度方程式，進而有效的增進求解的效率。為了數值的準確性，本論文提出了一個多維度的頻散關係得以保持之上風格式。經由測試問題的驗證，本論文所提出的求解方式能夠有效的求解高雷諾數及高瑞里數問題，它較前人所提出的方法更為有效並能減少計算時間。
而為了能夠求解複雜外型的流動問題，本論文也提出了一搭配頻散關係得以保持之雙緊緻格式的沉浸邊界方法。沉浸邊界方法的概念是在動量方程式中加入一虛擬源項，以其能達到滿足物體非滑移之邊界條件。本論文所提出之沉浸邊界方法，是經由求解微分方程式來達成插值的目的，故能較為容易的擴展至三維的模擬。經由測試問題可以得知，本論文所提出的沉浸邊界方法求解方法，與前人所模擬之結果均相當的吻合。
最後，本論文也將不可壓縮流體之求解器搭配MPI平行架構來做求解。藉由使用區域分割法及所提出的無散度補償方法，本論文所提出之平行架構能夠達成相當好的平行效能。對於拉穴問題，本論文所提出的平行方法加速比及效能比均具相當好的結果。

關鍵字

頻散關係得以保持 ； 非線性淺水波 ； 光電電磁波 ； 不可壓縮流 ； 沉浸邊界法

並列摘要

In this dissertation, schemes which accommodate a better dispersion relation for the convective terms shown in the transport equation, are proposed to enhance the convective stability by virtue of the increased dispersive accuracy.
All the schemes developed within this dissertation have been rigorously developed through the dispersion and dissipation analyses. To verify the proposed method, several problems that are amenable to the exact and benchmark solutions will be investigated. The results with good rates of convergence are demonstrated for all the investigated problems.
The dispersion-relation-preserving schemes are then employed to nonlinear shallow water wave equation, electromagnetic wave equation, and incompressible fluid flow equation. For the nonlinear shallow water wave equation, we develop a computationally efficient and spatially accurate iterative scheme for solving the nonlinear shallow water wave equation, called Camassa-Holm (CH) equation. For computing an accurate dispersive solutions of the equation, we cast the equation in terms of an auxiliary variable to yield a linear convection-reaction equation that is coupled with a Helmholtz equation.
We propose a sixth-order accurate advective scheme
which accommodates the dispersion relation for the first-order spatial derivative term, while solving the coupled Helmholtz equation by a sixth-order accurate compact scheme
formulated within a three-point stencil. In order to retain the Hamiltonian structure, a second-order-accurate and sixth-order-accurate symplectic time-stepping scheme is employed for the time integrator. Several test problems are provided, including a traveling wave solution, to justify the integrity of the proposed symplecticity- and
dispersion-relation-preserving discretization scheme. Comparisons between the proposed scheme, a complete integrable particle method, and the local discontinuous Galerkin method are also made in terms of accuracy,
the elapsed computing time, and spatial and temporal rates of convergence.
For the electromagnetic wave equation, a solver for electromagnetic wave equation, called Maxwell's equations,
is proposed in two- and three-dimensional non-staggered grids. To avoid even-odd spurious oscillations, the first-order spatial derivative terms will be approximated by the explicit compact scheme to save the computational time. To accommodate the Hamiltonian structure in the Maxwell's equations, the time integrator employed in the current semi-discretization falls into the symplectic category. The integrity of the finite difference time domain method for solving the TM-mode Maxwell's equations has been analytically verified through two- and three-dimensional test problems.
For the incompressible fluid flow equation, we aim to develop a new formulation to effectively calculate the solutions of incompressible fluid flow equation, called Navier-Stokes equations, in non-staggered grids. The distinguished feature of the proposed method, which avoids directly solving the divergence-free equation, is to add a rigorously derived source term to the momentum equation to ensure satisfaction of the fluid incompressibility. For the sake of numerical accuracy, dispersion-relation-preserving upwind scheme developed within the two-dimensional context was employed to approximate the convection terms. The validity of the proposed mass-preserving Navier-Stokes method is justified by solving two benchmark problems at high Reynolds and Rayleigh numbers. Based on the simulated Navier-Stokes solutions,
the proposed formulation is shown to outperform the onventional segregated method in terms of the reduction of CPU time.
In order to solve the flow equations in irregular and time-varying domains, a immersed boundary (IB) method developed in Cartesian grids is applied together with the dispersion-relation-preserving dual-compact scheme. The artificial momentum forcing term applied at certain points in cells containing fluid and solid allows an imposition of velocity condition to account for the motion of solid body. We develop in this study a differential-based interpolation scheme which can be easily extended to three-dimensional simulation. The results simulated from the proposed immersed boundary method agree well with other numerical and experimental results for the chosen benchmark problems. The accuracy and fidelity of the IB flow solver developed to predict flows with irregular boundaries are therefore demonstrated.
Finally, we employ the present incompressible Naver-Stokes equations and run the code in message passing interface (MPI) parallel platform. With the domain decomposition methods combined with our divergence-free-condition compensated method, we can easily get higher performance for the present framework. The speed-up and efficiency are good for the simulation of lid-driven cavity problem.

並列關鍵字

dispersion-relation-preserving ； nonlinear swallow water wave ； electromagnetic wave ； incompressible flow ； immersed boundary method

參考文獻

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