Acknowledgements

I would like to express my deepest appreciation to professor Bao Weizhu for his inspiration, guidance and encouragement. He also gave me the chance and lead me to the world of computational mathematics. Without his help, this thesis and many others would not be possible. I would like to thank Prof Pierre Degond and Prof Wang Zhisong for their support and guidance during the collaboration. They provide constructive suggestions along my research. I would like to thank my research fellows Ngoc and Hou Ruizheng for their discussion and inspirations. Special thanks are given to Zhao Xiaofei for helping me check the thesis. I would also like to thank friends Yuan Zihong, Huang Mengmin, Jia Xiaowei, Wang Yan and many others for their friendship and support. Finally I would like to thank my parents and my wife for their understanding, patience and love during the past several years.