The Research On The Construction And Properties Of WENO-AO-Type High Resolution Schemes

It is important to construct the high-resolution numerical scheme for the hyperbolic conservation laws equations,which usually have discoutinuous solution.For example,the entropy stable scheme is constructed for the physically solution,which satisfies the disrete entropy inequalities.And the well-balanced scheme is constructed for the steady-state perturbation problem.The sign-preserving is very important for the conservative variables before and after reconstruction during the construction process of a high-resolution entropy stable scheme.In the view of this,the sign-preserving property of WENO-AO(Weight Essentially Non-oscillatory with Adaptive Order)reconstruction is studied theroretically.And the two excellent schemes(WENO-AO-type entropy stable scheme and WENO-AO-type central upwind scheme)are constructed.The numerical results are compared by several examples.Moreover,the well-balanced property of the WENO-AO-type scheme,which shows better in several cases,are further studied.The property is proved theoretically and is tested by some cases.The specific work is as follows:(1)A new sign-preserving WENO-AO-type reconstruction is proposed and is combined with the corresponding numerical fluxes to obtain two types of high-resolution numerical schemes for hyperbolic conservation laws equations.The suitable reconstruction of variables is an important part of constructing high order numerical schemes,especially for entropy stable schemes,which impose restrictions on the reconstruction method.It is required that the reconstruction keeps same sign for the jumping at interfaces.Considering the good sign-preservation and stability of third-order compact CWENO reconstruction,this reconstruction is introduced to improve the original WENO-AO reconstruction and a new type of WENO-AO reconstruction is obtained,which is proved to be sign-preserving.Based on the sign-preserving WENO-AO-type reconstruction,two types of high-order numerical schemes were constructed.Firstly,the numerical flux adopts fourth-order entropy conservative flux,and the 5-order sign-preserving WENO-AO-type reconstruction for entropy variable in dissipation term.A class of fourth-order and sign-preserving entropy stable schemes is obtained.Secondly,by combining the new reconstruction with the central upwind numerical flux,a fifth order low dissipation central upwind scheme is obtained.Finally,two types of schemes are applied to specific examples to test the performance of different schemes.(2)The well-balanced preserving property of the high order sign preserving entropy stable scheme for shallow water equations with bottom topographic source terms is proved.Keeping well-balanced is an important property of numerical schemes for accurately characterizing steady-state problems.To accurately solve this problem,the constructed numerical schemes must satisfy well-balanced property,i.e.,the numerical flux and source terms must be balanced.To constructed the well-balanced scheme,based on the entropy conservative flux form of the shallow water equation,a linear combination of lower order discrete forms is used as the source term discretization method to accurately balance non-zero flux and source term.Furthermore,the well-balanced preserving theorems are proposed and are proved theoretically for the high order entropy conservative scheme and for the high order entropy stable scheme.Finally,This well-balanced property is fully verified numerically by solving some steady-state problems with small perturbations.