Research On High Order Accurate And High Resolution Difference Methods Of Fluid Dynamics

In computational fluid dynamics, the numerical simulation of flow with both discontinuities andcomplex structures is an urge and difficult task. To solve these problems, high-order high-resolutionnumerical methods have been proposed recently. In the numerical methods of fluid dynamics, finitedifference schemes are simple and easy to achieve high order, therefore they have been extensivelystudied. The main object of this thesis is to find difference schemes which are more suitable for aboveproblems. Besides that, there are some discussions about the convergence order and the stability ofdifference schemes. The main content includes the following parts:1. Research about the convergence order and the smoothness indicator of WENO schemesA sufficient condition for five-points WENO schemes achieving the fifth-order is given which iseasy to use. And the formula of accuracy orders of WENO schemes is also given. Furthermore, thesmoothness indicator of Jiang and Shu is improved for better resolution on complex structure. Inaddition to this, the first smoothness indicator proposed by Liu et al. is analyzed in detail.2. Research about compact schemes and hybrid compact WENO schemesA fifth-order conservative upwind compact schemes and an explicit boundary scheme whichmaintains the global fifth-order accuracy are given. Furthermore, based on the fifth-order hybridcompact WENO scheme proposed by Ren et al., a new treatment for Euler equations is given. Sincethis treatment has avoided the block tridiagonal equations, it reduces the computation costs greatly. Atthe same time, it increases the dissipation slightly because of its global flux splitting.3. Research about stability criterion of semi-discrete difference schemesThe stability condition for semi-discrete difference schemes obtained from Fourier analysis isusually an inequality of trigonometric function which is not convenient to use. For difference schemeon uniform grids, this trigonometric function can be converted into a polynomial form. Taking theconvergence order of the scheme into consideration, this polynomial can be factorized into a simpleform. Thus, the corresponding inequality is much easier to solve than the original one. This method isvery effective for judging the stability of semi-discrete difference schemes. Particularly, by using thismethod, it is practicable to add the stability constraints when designing high-order schemes.4. Research about optimized scheme and optimized WENO schemesA maximum order preserving optimized (MOPO) scheme which is an weighted average of themaximum order scheme and an optimized scheme is given. To determine the weights of these two linear weights, a local wavenumber indicator is introduced. Furthermore, a maximum order preservingoptimized WENO (MOPOWENO) scheme is achieved by integrating it with the sixth-points WENOschemes framework. Numerical results show that the proposed schemes combined both advantages ofthe maximum order schemes and the optimized schemes.