Riemannproblem

The Riemann problem is a fundamental concept in the field of computational fluid dynamics and hyperbolic partial differential equations. It was first introduced by Bernhard Riemann in his 1860 paper "On the Motion of Fluids." The problem involves the study of the propagation of discontinuities in the solution of hyperbolic conservation laws. These discontinuities, known as shocks, are regions where the solution changes abruptly.

The Riemann problem is typically formulated as an initial value problem for a system of hyperbolic conservation laws, where the initial data consists of a single discontinuity, or "Riemann jump," in the solution. The goal is to determine the structure of the solution as it evolves in time. This involves identifying the waves that propagate from the discontinuity and their interactions.

The Riemann problem is important because it provides a simplified model for studying the behavior of shocks and other discontinuities in more complex problems. It has been extensively studied and has led to the development of various numerical methods for solving hyperbolic conservation laws, such as the Godunov method and the Roe method. These methods are widely used in computational fluid dynamics and other fields where hyperbolic conservation laws arise.