"大气数值模拟理论与方法课程-数值追踪问题解析:分类与时间差分法"

The document "Lecture 8: Numerical schemes for advection problems" is a part of the graduate level course on Atmospheric Modeling. It covers the theoretical foundations and methods for atmospheric numerical simulations, with a specific focus on numerical schemes for advection problems. The document begins with a discussion on the classification of partial differential equations (PDEs), emphasizing the distinction between hyperbolic, parabolic, and elliptic equations. It then delves into the time differencing methods for solving PDEs, highlighting the significance of explicit schemes for the advection equation. The advection equation is presented in its general form, and the document goes on to provide insights into the second order linear PDE, distinguishing between hyperbolic, parabolic, and elliptic forms. This discussion sets the stage for understanding the numerical schemes for advection problems. The document also presents the simplest examples of PDEs, including the wave equation, diffusion equation, and Laplace's equation. Each example is explained in the context of advection problems, providing a comprehensive foundation for students to grasp the numerical methods required for atmospheric modeling. Overall, the document serves as a valuable resource for graduate students in Atmospheric Modeling, equipping them with the necessary theoretical knowledge and numerical methods to tackle advection problems in atmospheric simulations. It emphasizes the importance of understanding the classification of PDEs and time differencing methods, and provides a solid framework for students to build their expertise in numerical schemes for advection problems.