A higher order numerical method for transonic flows

Abstract

This thesis presents and verifies a numerical method for solving the compressible Euler equations. The method is based on a finite volume method with an upwind type TVD dissipation terms originally developed by Harten ( 1983 ) for scalar hyperbolic conservation law and extended to Euler equations by using Roe's approximate Riemann solver. The present method has second-order accurate in smooth region of the solution and intelligently switches the scheme to first-order accurate in the vicinity of shocks to presents a sharp and smooth shock wave profile. The present method contains no user-dependent and problem-dependent parameters. An explicit multistage Runge-Kutta time stepping is used to integrate the system. A multigrid method is employed in the present method to accelerate to convergence. Meanwhile a fully implicit time integration scheme is also investigated and adopted in this method. The explicit multistage time stepping with the multigrid acceleration is modified to solve the fully implicit system. The present method is programmed in two-dimensions for the Euler equations aiming at the application to internal flows. Numerical experiments are carried out to test the accuracy and the efficiency of the present method. Results compare well with exact solutions and perform better than some well-documented results. The desired efficiency is obtained. The connection between central difference and upwind difference is investigated. It is found that the widely used Jameson's central differencing plus explicit adaptive artificial viscosity can be interpreted as a hybrid scheme by a weighted average of a first order upwind scheme and a second order upwind scheme.