Abstract:
The advection equation is considered and solved using methods such as the upwind method,
Lax-Wendorff method, minmod limiter, MC limiter, and super bee limiter. The outcomes from
these methods are then compared with the exact solution, and the results are presented. The
findings indicate that the upwind method requires more mesh points, whereas the superbee and
MC limiter outperform the minmod limiter. The Burger’s equation is subsequently evaluated
using numerical approaches, specifically the Roe approximation and Godunov’s method. After
comparing the outcomes with the exact solution, it was determined that Godunov’s method
produces results closely aligning with the exact solution. Thus, Godunov’s method is the pre ferred choice. After that, the compressible Navier-Stokes equation is examined using a test case
involving supersonic flow past a cylinder. Contour plots have been created for both velocity
components and density. The findings indicate that with more mesh points, the skin friction
coefficient improves, though it’s not entirely accurate. Subsequently, the Euler equation was
considered and studied the supersonic flow over a circular cylinder. The advection equation is revisited, and the discontinuous Galerkin method is utilized for spatial discretization, while the RK
method is employed for time discretization. We examined the stability of various RK methods.
