Research On Space And Time Discrete Method Based On Flux Reconstruction Method

Hyperbolic conservation law equations are one of the most important types of governing equations in computational fluid dynamics.The numerical solution method is not only the focus of the research on the numerical solution of partial differential equations,but also the difficulty.We can usually only get the weak solution of the equation,so we need to limit it to get a solution that is consistent with the physical background.The limitation method mainly studies the problem from two aspects: one is from the perspective of energy stability,and the other is from the perspective of entropy stability.Among them,the energy-stable format has a simpler structure and higher accuracy,and has attracted the attention of many scholars in recent years.The flux reconstruction idea is the core of the format method.Based on this format,many scholars have proposed various flux reconstruction methods.Because the original flux reconstruction method has obvious defects in accuracy and stability,Huynh proposed a highorder flux reconstruction method based on previous work.The essence of this method is to divide the flux into discontinuous flux and Correct the flux,and perform high-order reconstruction on this basis to obtain the flux at the interface of the numerical unit.In this paper,based on Huynh's method,Nodal Discontinuous Galerkin method and Spectral Difference method are combined in the proof of high-order modified flux structure and energy stability,thereby obtaining a new format based on the flux reconstruction method,namely high-order energy stability.format.This method has a clear physical background,does not need to add artificial dissipation terms in the calculation,and has high accuracy,can effectively avoid the generation of non-physical understanding,and can effectively solve the hyperbolic conservation law equations.At present,the high-order methods of flux reconstruction methods mainly revolve around high-precision spatial discretization methods.This paper is the first time to study the timediscrete format based on the flux reconstruction algorithm,and compares the 3rd-order RungeKutta method with the precise integration method proposed by Academician Zhong Wanxie.The flux reconstruction algorithm(FR)usually adopts the third-order Runge-Kutta method or the fourth-order Runge-Kutta method in the time discrete method.In order to match the time discrete method with the high-precision spatial discrete method,this paper selects the precise integration method for research.The semi-discrete equations obtained by the flux reconstruction method are in the form of non-homogeneous equations.In this paper,the increment-dimensional method is used to transform the non-homogeneous differential equations into homogeneous differential equations.It is unnecessary to perform matrix inversion in the process of implementing precise integration,which greatly reduces.The amount of calculation is conducive to the realization of programming.Comparing the results of the Runge-Kutta method and the precise integration method,the feasibility of applying the increment-dimensional precise integration method to the flux reconstruction algorithm is verified.