A One-Step Lie-Group Shooting Method on Solving Non-Linear Multi-Dimensional Backward Burgers’ Equation with High Reynolds Number for Long Time Span

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基本資料

Project title

Code/計畫編號

MOST109-2221-E019-015

Translated Name/計畫中文名

一步李群打靶法求解具高雷諾數多維非線性時間反向Burgers問題

Project Coordinator/計畫主持人

Yung-Wei Chen

Funding Organization/主管機關

National Science and Technology Council

Department/Unit

Department of Marine Engineering

Website

https://www.grb.gov.tw/search/planDetail?id=13534391

Year

2020

Start date/計畫起

01-08-2020

Expected Completion/計畫迄

31-07-2021

Bugetid/研究經費

655千元

ResearchField/研究領域

機械工程

本計劃擬提出完整一步李群打靶法求解長時間、非線性、不連續、多維度時間反向Burgers問題。對於Burgers方程正算問題考慮低動黏滯係數將發生不連續和高梯度問題，利用傳統方法級數展開無法有效克服發散與數值不穩定；對於反算問題，直至目前為止高雷諾數與長時間跨距影響下，上述極端問題對於時間反向計算尚未有文獻提出與有效計算。若反算問題解存在，會高度依賴給定的資料，特別是當量測資料有微小擾動時，將造成很大的震盪在於求解上。因數值方法求解過程將產生解唯一性問題。本計畫首先利用李群特性推導初值-終值兩點邊值解，透過李群解在歐式與閩式空間存在的唯一性，引入邊界條件避免時間積分與不連續問題，進一步利用非線性、高雷諾數、高維度、及長時間計算進行數值驗證，實現高雷諾數下一步保群法之數值實驗，研究成果將率先突破國際反問題求解程序必須利用迭代進行求得穩定解。由於非線性對流項與擴散項將形成不連續和高梯度問題，對於傳統迭代型式李群打靶法將產生多解問題。本計畫根據李群元素G(tf)與廣義值原理G(r)進行建構李群打靶法。根據李群相似轉移矩陣G(tf)=G(r)，透過單參數r進行求解待求初始條件。因單參數G(r)進行求解時初始與終止條件不符合保群算法:光錐構造、李代數、群的特性將發生發散及零解，將無法符合一步保群基本理論。因此，引入初始與終止限制條件避免求解過程的發散。根據李群特性必存在唯一解，利用閩式空間下運動方程建構一元二次方程式推導初始與終止條件互易性關係。利用兩點邊界條件求得每時刻下特徵線變化率，以避免直接積分不連續與高梯度產生多解與發散現象。本計畫分別針對非線性、高雷諾數、多維度、及長時間條件下之數值算例測試。此嚴苛計算問題直到目前尚未被挑戰及提出克服方法。一步李群打靶法之數值精度、效率及噪音影響將被驗證與先前參考文獻比較。This project will develop the completely one-step Lie-group shooting method for solving nonlinear, large Reynolds number, multi- dimensional backward Burgers’ equation under long time span. For this problem included nonlinear convective term and diffusive term, there can be velocity discontinuities and high gradient. It is very difficult to deal with numerical divergence and to avoid numerical instability by conventional methods. Up to date, none of literatures have not been proposed to address for the large Reynolds number and the long-time span in backward in time of inverse problems. If a solution is existence for an inverse problem, the solution of the inverse problem is highly dependent on the input data. Small perturbations in the input data, like random errors inherent to the measurements in the analysis, can cause large oscillations on the solution. The numerical procedure of traditional schemes will occur multiple solutions problem when considering nonhomogeneous boundary conditions and a large Reynolds number. According to the Lie group properties, the solution of the two-points boundary conditions can be derived and proved the unique solution in Euclidean space and Minkowski space. We will use numerical experiments to verify the theory of one-step Lie group shooting method (LGSM), which can obtain efficient and stable solution, and do not need to numerical iteration. To date, there has been no open report that shows the numerical methods can calculate this severely ill-posed multi- dimensional backward Burgers’ equation under long time span.Because nonlinear convective term and diffusive term arise in engineers, velocity discontinuities and high gradient problems must occur. The LGSM is based on a one-step Lie group element G(tf) and a generalized mid-point Lie-group element G(r). Then, according to G(tf) = G(r), we can search for the missing initial conditions through the weighting factor. The main reasons are that a real single-parameter G(tf) occurs three conditions, zero, infinite and complex number, and the GPS cannot preserve above three constrain in time direction. Therefore, a constrain condition of initial condition (IC), final condition (FC) and weighting factor is conducted and avoided the divergence of the solution. According to the quadratic equation of the LGSM, a solution is applied to obtain the IC and to examine the FC. Using the reciprocal relationship of the solution of forward and backward schemes, the proposed algorithm can avoid time integration of the numerical scheme and the numerical divergence. To illustrate the effectiveness and accuracy of the proposed algorithm, several benchmarks in multi-dimensions are tested.

Keyword(s)

高雷諾數
非線性
逆問題
病態問題
迭代法
High Reynolds number
Non-linear
Inverse problem
Ill-posed problem
Iterative regularization method

DSpace CRIS

A One-Step Lie-Group Shooting Method on Solving Non-Linear Multi-Dimensional Backward Burgers’ Equation with High Reynolds Number for Long Time Span

基本資料

Description