Analytical Solutions And Semi-analytical Numerical Methods For Burgers' Equation

Burgers' equation is an important non-linear equation. It is the preliminary model to describe the interaction between convection and diffusion. The equation has many important characteristics, such as the laws of conservation, mixed partial differential equations, nonlinearity and so on, which leads the high value of research and application. However, due to the appearance of the shock wave, the development of a robust numerical method for solving Burgers' equation with large Reynolds number accurately and efficiently, remains as a challenging task.Firstly, this thesis gives analytical solutions for two and three dimensional Burgers' equation with the special initial condition. When the initial condition is the special combination of sine and cosine functions, the solution of Burgers' equation can present shock wave phenomenon, and up to now, there is no analytical solution for this problems in references. Based on the Hopf-Cole transformation and separate variables method, the solution is given in the form of infinite series. But, for computing each term, double integral or triple integral must be taken into account, and these integrands consist of the exponential function associated with the Reynolds number and the trigonometric function related to the series terms. With the increasing of Reynolds number, the efficiency of calculating the analytical solution is quit slow. Thus, in this thesis, the double integral and the triple integral are simplified into a function composed of Bessel function, hypergeometric function, power function and factorial. Using this simplification, the computational efficiency of the analytical solutions is improved significantly. The two analytical solutions for the special problems provide benchmarks for testing the accuracy of numerical methods.Secondly, in this thesis, a semi-analytical numerical method is proposed for solving one and two dimensional Burgers' equation involving a large Reynolds number. Based on the Hopf-Cole transformation and symmetry transformation, the one and two dimensional Burgers' equation in finite-space domain is transformed into heat-transfer equation in the infinite space domain. By using Green function method, the analytical solution for the heat transfer equation is obtained, then the solution of Burgers' equation can also be obtained. However, using this solution directly face two problems, i.e. the precision problem in floating point calculation and the numerical integration in the infinite domain. Thus, in this thesis, based on a rigorous error analysis, the integration in infinite domain is transformed into the integration in finite domain, which resolves the numerical difficulties. The numerical examples indicate that:the semi-analytical numerical method proposed in the thesis performs perfectly in many aspects, involving accuracy, efficiency, stability as well as reliability, and the ability of capturing shock wave. For larger Reynolds numbers, when the other numerical methods fail completely, the proposed method can give accurate and stable results.