Numerical Solution of Two-Dimensional Inviscid Euler Equations

Solving the two-dimensional inviscid Euler equations presents a fascinating mathematical challenge with broad applications in gas dynamics and fluid mechanics. This problem requires developing robust numerical methods to obtain practical solutions. The specific scenario involves flow through a 2D converging-diverging nozzle, which has significant real-world engineering relevance. Implementation typically involves discretizing the governing equations using finite-volume methods and employing flux calculation schemes like Roe's approximate Riemann solver or the Steger-Warming flux splitting method. Key computational aspects include handling boundary conditions for inlet pressure/temperature and outlet pressure parameters, implementing time-marching algorithms (explicit or implicit), and ensuring conservation of mass, momentum, and energy through appropriate flux discretization. While computationally demanding, structured grid generation coupled with shock-capturing techniques enables accurate simulation of compressible flow behavior including potential shock formations within the nozzle geometry.