A PDE, or partial differential equation, is a type of mathematical equation that involves multiple variables and their partial derivatives. These equations are used to model physical phenomena in fields such as physics, engineering, and economics.
In the context of PDEs, characteristics refer to the curves along which the solution to the equation remains constant. These curves are important in solving PDEs as they help determine the behavior of the solution.
Expansion waves are a type of shock wave that occurs when a disturbance travels through a medium at a supersonic speed. In the context of PDEs, expansion waves can occur in equations that involve non-linear terms, and they can be used to analyze the behavior of the solution.
Shocks, also known as discontinuities, are sudden changes in the solution of a PDE. They can occur when the characteristics of the equation intersect, leading to a breakdown in the smoothness of the solution. While expansion waves are a type of shock, shocks can also occur in other forms, such as rarefaction waves and contact discontinuities.
Some common methods for solving PDEs involving characteristics, expansion waves, and shocks include the method of characteristics, the shock-fitting method, and the Godunov method. These methods involve breaking down the PDE into simpler parts and using numerical techniques to approximate the solution.