Hybrid WENO Methods For Hyperbolic Conservation Laws

The main work of this paper is to design a set of finite difference weighted essentially non-oscillatory(WENO)methods for steady and unsteady problems in fluid mechanics.This kind of method not only maintains the optimal convergence in the smooth flow field,but also keeps essentially non-oscillatory(ENO)property in the strongly discontinuous region.What’s more,compared with classical WENO methods,our methods possess lower numerical dispersion and dissipation errors,could capture more abundant flow field details.This method is mainly developed to solve unsteady and steady Euler equations in this paper.We designed a new smoothness detector to identify smooth region in the flow field,and introduced the high-frequency region to capture high-frequency wave effectively.For the steady problem,we introduced the transition region,and utilized unequal-sized stencils to construct the reconstruction polynomial in the discontinuous region,achieving robust steady-state convergence.There are mainly three following parts in this thesis:In the first part,we studied the finite difference method(FDM)for hyperbolic conservation laws,and designed a new hybrid WENO scheme.In this part of the thesis,our main innovation points are divided into two parts.The first major innovation is that we designed a new smoothness detector,we divided the computational domain into three parts:smooth region,high-frequency region and discontinuous region.The detector can accurately identify the smooth region in the flow field,in this region,we utilized the optimal linear reconstruction to ensure the optimal convergence accuracy of the numerical scheme for the smooth problems.In the discontinuous region,we utilized the classical WENO reconstruction for shunting,thus effectively suppressing the numerical oscillation for the strong discontinuity problem.The second innovation is the hybrid reconstruction method used in high-frequency region.We made a convex combination of the optimal linear reconstruction and the classical WENO reconstruction to obtain a new mixed polynomial in the high-frequency region,and properly control the smoothness of the mixed polynomial,thus maintaining the high-resolution advantage of the linear reconstruction and inheriting the essentially non-oscillatory property of WENO reconstruction.Spectral analysis and lots of benchmark examples demonstrated that the new hybrid WENO scheme possesses lower numerical dispersion and dissipation errors than the classical WENO schemes.In the second part,we studied the steady-state convergence problem of steady Euler equations on a rectangular computational domain.We modified the hybrid reconstruction method proposed in the first work properly,and the new scheme can make the residual error of the steady Euler equations converge to the machine error level.We still utilized the smoothness detector proposed in the first work to divide the computational domain into three parts:smooth region,transition region and discontinuous region.In the smooth region,the optimal linear reconstruction is used to maintain the optimal convergence accuracy.We adopted unequal-sized stencils to construct WENO reconstruction polynomial,and made a convex combination of its corresponding linear scheme and it in the discontinuous region,so that the scheme can maintain high resolution near the discontinuity and achieve robust steady-state convergence.In the transition region,the smoothness detector is used as the medium,and the continuous transition between the smooth region and the discontinuous region is achieved through interpolation technique.Numerical experiments showed that the setting of the transition region plays an important role in the steady-state convergence of the hybrid scheme.Many benchmark examples illustrated that the new hybrid scheme possesses more robust steady-state convergence property than the classical central weighted essentially non-oscillatory(CWENO)scheme and the multi-resolution weighted essentially nonoscillatory(MRWENO)scheme,and shows a better ENO property near strong discontinuities.In addition,spectral analysis and a large number of numerical examples also demonstrated that the new scheme has lower numerical dispersion and dissipation errors.In the third part,we studied the steady-state convergence of steady Euler equations with general curved boundaries,which is still computed on Cartesian grids in the framework of finite difference.In this situation,the robustness of the steady-state convergence of the classical fifth-order CWENO and MRWENO schemes is somewhat weakened.The experimental results showed that proper boundary treatment has a great effect on the robustness of the steady-state convergence of the whole scheme.We designed a series of extrapolation methods and used the technique of inverse Lax Wendroff(ILW)to deal with the curved boundary.We utilized numerical differentiation and the information of curvature radius to transform the tangential derivative in the control equation,respectively.Both of them successfully achieved robust steady-state convergence for the series of benchmark examples.Compared with the work with MRWENO as the internal scheme,we increased the power of the weighted combination in the extrapolation methods from one to three.Numerical analysis showed that this manner can ensure that the extrapolation methods provide fifth-order approximation accuracy near the boundary.In addition,we optimized the process of stencil selection for two-dimensional extrapolation,making it more suitable for the simulation of general problems.