Cartesian grid and Discontinuous Galerkin Methods for Conservation Laws

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.