Numerical Simulations of the Steady Euler Equations on Unstructured Grids

This thesis is concerned with effective and robust numerical schemes for solving steady Euler equations. For solving the nonlinear system resulting from the discretization of the steady Euler equations, we employ a standard Newton method as the outer iterative scheme and a linear multigrid method as the inner iterative scheme with the block lower-upper symmetric Gauss-Seidel iteration as its smoother. The Jacobian matrix of the Newton-iteration is regularized by the local residual, instead of using the commonly adopted time-stepping relaxation technique based on the local CFL number. The local Jacobian matrix of the numerical fluxes are computed by using the numerical differentiation, which can significantly simplify the implementations by comparing with the manually derived approximate derivatives. In the reconstruction step, the linear reconstruction and the quadratic reconstruction are studied respectively. For the linear reconstruction, the approximate polynomial in each cell is obtained by using the WENO reconstruction method. The numerical results show that the algorithm works very well with the WENO reconstruction. Compared with the resutls given by using the Venkatakrishnan limiter, the WENO reconstruction method gives superior convergence order, and nonoscillatory and sharp shock profiles. Although the WENO method works very well for the linear case, the convergence to the steady state of the algorithm is affected if the WENO method is extended to the quadratic case directly. So for the quadratic reconstruction, a new hierarchical WENO reconstruction method is introduced to improve the convergence to steady state and also to preserve the formal order of accuracy. The efforts are made to balance the convergence order of the numerical