Construction Of Several High Precision And High Resolution Schemes With Their Applications

The numerical solution of the hyperbolic conservation laws equation is a very important method in the computational fluid dynamics.Among the endless nu-merical methods,high precision and high resolution numerical methods play an important role in the development of computational fluid dynamics because of their physical characteristics.The main purpose of this paper is to study several numer-ical schemes with high resolution and precision.The details are as follows:Firstly,based on the idea of finite volume method,we improved the classi-cal CWENO-Z scheme.By increasing the weight of the non-smooth part of s-moothness factor,we proposed the third-order CWENO-Z3+scheme and the fourth-order CWENO-Z4+scheme respectively,which both can improve the resolution of CWENO-Z scheme.By discussing the accuracy of CWENO-Z scheme at the critical points,we deduce it by Taylor expansion theory.The two-parameter CWENO-NZ3 scheme and CWENO-NZ4 scheme can effectively improve the accuracy of the scheme at the critical point.Comparisons of numerical experiments show that the four new CWENO schemes not only improve the calculation accuracy at the critical points,but also reduce the dissipation of the schemes,thus improving the resolution of the flow field structure.For the third-order and the fourth-order central-upwind schemes,severe non-physical oscillations will occur in some fluid problems calculations,which will lead the calculation failure.In order to solve this kind of problem,we introduce two-point Gauss polynomials to modify the original central scheme locally,and establish a new Gauss-type central-upwind scheme,namely GCWENO scheme,which not only keeps high precision and high resolution,but also has simple structure and can effectively eliminate the non-physical oscillation of the central-upwind scheme.By using this scheme,we have solved a large number of one-dimensional and two-dimensional problems with strong discontinuity and instability.Comparisons of numerical results show that the GCWENO scheme can effectively eliminate the numerical oscillation of the original scheme,and has good stability in calculating strong discontinuous problems such as RT instability.NND scheme is a kind of high resolution finite difference scheme based on the requirement of smooth upstream and downstream shocks without oscillation.However,the resolution of NND scheme will become lower when calculating complex flow due to the minmod limiter.In order to improve the resolution of the scheme,we combine the flux splitting technique in the finite difference scheme with the idea of second-order MUSCL interpolation reconstruction.By restricting the flux,we develop a kind of mixed difference schemes with high resolution,namely MNS scheme.MNS scheme will degenerate into MUSCL scheme when calculating linear problems.Therefore,MNS scheme is a TVD scheme.And it will degenerate into NND scheme when minmod limiter is used,so it also has the high resolution property of NND scheme.Compared with the classical MUSCL scheme and NND scheme,the MNS scheme is proved to be a higher resolution TVD scheme.Numerical oscillation occurred in the numerical experiments,which means cal-culation failure,when we use the MUSCL scheme and the MNS scheme constructed in this paper to calculate some complex flow with superbee limiter.Considering the insufficient resolution of the minmod limiter and the instability of the superbee limiter in the calculation process,we combine the minmod,MC and superbee lim-iters by introducing a MAX function,and develop a generalized Roe-Sweby limiter with good stability.Using this idea,we also improved the van Albada limiter and van Leer limiter to construct a symmetric generalized van Albada limiter and a generalized van Leer limiter with high resolution and monotonicity.Through the-oretical analysis and numerical calculation,it is found that the three new limiters are efficient limiters with high resolution,and they have good ability to capture discontinuous solutions and sparse waves.