Solving One-Dimensional Euler Equations Using Finite Volume Method with Second-Order MUSCL Upwind Scheme

The finite volume method is a widely used numerical approach for solving one-dimensional Euler equations. In our simulation of isentropic nozzle flow, we implement a second-order MUSCL upwind scheme that effectively handles normal shock waves appearing in the divergent section of the nozzle. The MUSCL scheme represents a high-precision numerical method that maintains computational accuracy while achieving efficient calculation speed through slope limiting and reconstruction techniques. Implementation details include: - Cell-wise primitive variable reconstruction using minmod limiter to prevent oscillations - Riemann solver integration at cell interfaces for flux computation - Time integration via explicit methods like Runge-Kutta schemes - Boundary condition treatment for nozzle inlet/outlet flows Understanding and implementing the MUSCL scheme is crucial for advancing research and applications in computational fluid dynamics, particularly for shock-capturing simulations. The method's TVD (Total Variation Diminishing) property ensures stable solutions even near discontinuities.