Newton-like methods for fast high resolution simulation of hypersonic viscous flows

Abstract

Two Newton-like methods, i.e. the sparse finite difference Newton method and the sparse quasi-Newton method, are applied to the Navier-Stokes solutions of hypersonic flows using the Osher flux difference splitting high resolution scheme. The resulting large block structured sparse linear system is solved by a new multilevel iterative solver, the α-GMRES method, which includes a preconditioner and a damping factor. The algorithm is demonstrated to provide fast, accurate solutions of the hypersonic flow over a cone at high angle of attack. Being parallelizable on distributed memory multiprocessors and having an ability to tackle non-linear problems, it has great promise in tackling more complex practical air vehicle configurations. As a by-product of using the GMRES method, in which Hessenberg matrices are generated, the eigenvalues of the linear system can be estimated using the Arnoldi method. The spectra produced provide some insight into the behaviour of the GMRES method for different linear systems corresponding to different preconditioning and damping.