We consider the problems of simulation and control for Burgers' equation with mixed boundary conditions. We first conduct numerical experiments to test the convergence and stability of two standard finite element schemes for various Robin boundary conditions and a variety of Reynolds numbers. These schemes are used to compute LQR feedback controllers for Burgers' equation with boundary control. Numerical studies of these feedback control laws are used to evaluate the performance and practicality of this approach to boundary control of non-linear systems. / Master of Science

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Sun Siu-wing. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2002. / Includes bibliographical references (leaves 50-51). / Abstracts in English and Chinese. / Acknowledgments --- p.i / Abstract --- p.ii / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Equation without viscosity --- p.5 / Chapter 3 --- Equation with standard viscosity --- p.8 / Chapter 3.1 --- "Particular convective flux, f(x) =u2" --- p.8 / Chapter 3.2 --- Convex convective flux --- p.10 / Chapter 4 --- Equation with monotonic dissipative flux --- p.11 / Chapter 4.1 --- Large initial data --- p.12 / Chapter 4.2 --- Small initial data --- p.19 / Chapter 4.3 --- Unbounded dissipative flux --- p.28 / Chapter 5 --- Equation with non-monotonic dissipative flux --- p.31 / Chapter 5.1 --- Large initial data --- p.32 / Chapter 5.2 --- Small initial data --- p.37 / Chapter 6 --- Comparison and conclusions --- p.39 / Appendices --- p.42 / Chapter A --- Hopf-Cole transformation --- p.42 / Chapter B --- Dirichlet problem --- p.45 / Bibliography --- p.50

一維的無粘Burgers方程是最簡單的非線性雙曲守恆型方程，在本篇論文中，我們提出一個交錯間斷伽遼金方法去解Burgers方程。交錯間斷伽遼金方法融合了標準有限元方法和標準間斷伽遼金方法，此方法會求兩個間斷函數的解，而這對函數間斷的地方是不同的，所以在其中一個函數間斷的位置，另外的函數加強了該函數的連續性。對於Burgers方程來說，要求的解及通量組成了一對交錯對，我們將構造這個交錯間斷伽遼金格式和證明這格式是能量守恆的。 / 典型Burgers方程的解常存有衝擊波和間斷的地方，在這些情況下，我們的格式不再是能量守恆，並且出現了數值振蕩的問題，我們會提出兩個方案去除掉數值解中的數值振蕩。第一個方法是把一個人工的擴散性通量加在數值格式裏，這個人工的擴散性通量是從一個解粘性Burgers方程的交錯間斷伽遼金格式中求得的，這個格式的構造過程跟構造原格式的過程是類似的。為確保數值解的準確度，擴散性通量只會在存有數值振蕩的地方才加上。第二個方法是一個全變差正則化方法，在某些保留數值解的準確性的條件下，振蕩性數值解的全變差會被減至最小。這個步驟只用於存在振蕩的地方，以減小計算成本和多餘的誤差。另外，處理最小化問題時會用到Bregman算法。本篇論文將記述有關這兩個方法的細節和數值驗証。 / The 1D inviscid Burgers' equation is the simplest nonlinear hyperbolic conservation law. In this thesis, a staggered discontinuous Galerkin method for the Burgers' equation is proposed. Staggered discontinuous Galerkin method is a kind of DG method that compromise conforming finite element method and standard DG method. Two unknown functions that are discontinuous at different points are solved, thus extra continuity is imposed at the points of discontinuity of the discontinuous function by the staggered counter part. For the Burgers' equation, the unknown function and the flux form the staggered pair. We will derive this staggered DG scheme and show that the scheme is energy conserving. / Typical problems concerning the Burgers' equation involve shock waves and discontinuous solutions. In such cases, the scheme is no longer energy conserving and the problem of numerical oscillations arises. Two approaches are presented to eliminate the numerical oscillations in the solution. The rst one is based on adding an artificial diffusive flux to the scheme. The artificial diffusive flux is derived from a staggered DG scheme for the viscid Burgers' equation for which the derivation is similar. To preserve accuracy, the artificial diffusive flux is added only at regions with oscillations. The second approach is a TV regularization method. The total variation of the oscillatory numerical solution is minimized under certain constraints that preserve the accuracy of the solution. To reduce computation cost and redundant error, the TV minimization process is induced locally in regions with oscillations. Bregman algorithm is applied for numerical implementation of the minimization problem. Detailed description of the two methods and the numerical results are presented in this thesis. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Chan, Hiu Ning. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 71-73). / Abstracts also in Chinese. / Chapter 1 --- Introduction --- p.7 / Chapter 2 --- Inviscid scheme --- p.10 / Chapter 2.1 --- Space discretization and element spaces --- p.10 / Chapter 2.2 --- Derivationofinviscidscheme --- p.11 / Chapter 2.3 --- Conservationofenergy --- p.13 / Chapter 2.4 --- Piecewiseconstantcase --- p.17 / Chapter 2.5 --- Problemwithdiscontinuity --- p.18 / Chapter 3 --- Mixed method --- p.21 / Chapter 3.1 --- Viscidscheme --- p.22 / Chapter 3.1.1 --- Derivation of viscid scheme --- p.22 / Chapter 3.1.2 --- Conservationofenergy --- p.24 / Chapter 3.1.3 --- Piecewiseconstantcase --- p.26 / Chapter 3.2 --- Relations between the inviscid scheme and the viscid scheme --- p.27 / Chapter 3.3 --- Mixed method with piecewise constant elements --- p.32 / Chapter 3.4 --- Mixed method with piecewise linear elements --- p.35 / Chapter 3.5 --- Numericalresults --- p.40 / Chapter 3.5.1 --- Figures --- p.40 / Chapter 3.5.2 --- Error --- p.49 / Chapter 4 --- A local TV regularization method --- p.56 / Chapter 4.1 --- LocalTVminimizationproblem --- p.56 / Chapter 4.2 --- Oscillationvector --- p.57 / Chapter 4.3 --- Methoddescription --- p.59 / Chapter 4.4 --- Implementation --- p.61 / Chapter 4.5 --- Remarkon’global’method --- p.63 / Chapter 4.6 --- Numericalresults --- p.63 / Chapter 5 --- Conclusion --- p.69 / Bibliography --- p.71

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Thesis (Master)--İzmir Institute Of Technology, İzmir, 2005 / Keywords: Vortex, burgers equation, dynamical systems, integrable systems, Euler equations. Includes bibliographical references (leaves. 92-97).

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Burgers’ Equation ut + cuux = νuxx is a nonlinear partial differential equation which arises in models of traffic and fluid flow. It is perhaps the simplest equation describing waves under the influence of diffusion. We consider the large time behavior of solutions with exponentially localized initial conditions, analyzing the rate of convergence to a known self similar single-hump solution. We use the Cole-Hopf Transformation to convert the problem into a heat equation problem with exponentially localized initial conditions. The solution to this problem converges to a Gaussian. We then find an optimal Gaussian approximation which is accurate to order t−2. Transforming back to Burgers’ Equation yields a solution accurate to order t−2.