High-order Accuracy WCNS Methods And Their Applications

High-order and high-resolution numerical methods for governing equations of fluid dynamics have become a decisive factor in the development of computational fluid dynamics(CFD).In this thesis,combining explicit and semi-implicit Runge Kutta time discretization methods,a series of explicit and semi-implicit high-order accuracy WCNS schemes are designed for solving pollution transport,steady hyperbolic conservation laws and stiff partial differential equations.High-order accuracy WCNS scheme is used for spatial discretization.In order to improve the computational efficiency,explicit and implicit time discretization methods are used for the non-stiff and stiff equations,respectively.The linear equations generated by the semi-implicit high-order accuracy WCNS scheme are solved by the GMRES algorithm based on Krylov subspace.The explicit and semi-implicit high-order accuracy WCNS scheme designed in this thesis is used to solve the following problems:For the pollution transportation equations with source terms,the source terms of the equations are split in order to keep the harmony of the stationary water solution(that is,the non-zero flux gradient and source terms are accurately balanced).The flux gradient and the spatial discretization in the source term are calculated with the fifth-order hybrid WCNS scheme,and the time discretization is calculated with the third-order explicit TVD Runge-Kutta method.Numerical experimental results show that the algorithm satisfies harmony under stationary water conditions,can obtain high accuracy in the smooth area,has good stability,keep high resolution and strong shock wave capture ability when simulating dam break problems.For steady hyperbolic conservation laws problem,the pseudo time derivative is introduced,and the third-order explicit TVD Runge Kutta method is used to calculate the time discretization,and the third-order explicit WCNS scheme is used to calculate the spatial discretization.In order to improve the computational efficiency,a fast-sweeping WCNS scheme with fast sweeping technique is designed.The core idea of the fast-sweeping method is to use the alternating scan sequence and Gauss Seidel type iterative method to solve the discretized nonlinear equations.Compared with the traditional fixed point iterative method,this method is not sweeping from a single direction,but from four directions.Numerical experiment results show that the fast-sweeping WCNS scheme has high accuracy.Compared with the explicit TVD Runge-Kutta WCNS scheme,it can reduce the number of iterations,reduce the CPU time,and has a strong shock wave capture ability.For the viscous Burgers equation,the viscous term is stiff.A third-order semi-implicit WCNS scheme is designed.The convective term and the viscous term are treated explicitly and implicitly,respectively.Compared with the third-order explicit TVD Runge-Kutta WCNS scheme whose time step is limited by the parabolic CFL stability conditions,the time step of the third-order semi-implicit WCNS scheme is only limited by the CFL stability conditions of the convection term.The fifth order explicit WCNS scheme is used to discretize the flux and the third order IMEX Runge Kutta method is used for time discretization.Through theoretical analysis,the stability conditions of the semi-implicit WCNS scheme are given.Numerical results show that the third-order semi-implicit WCNS scheme has high time accuracy,and has higher calculation efficiency than the third-order explicit WCNS scheme under the same conditions,and has a strong shock wave capture ability.For compressible Euler equations,the pressure term is stiff.The third-order semi-implicit WCNS scheme is designed,and the convection term and pressure term are treated explicitly and implicitly,respectively.The time step of the third-order semi-implicit WCNS scheme is only limited by the CFL stability condition of the convection term.Under the condition of low Mach number,it is more efficient than the third-order explicit TVD Runge Kutta WCNS scheme whose time step is limited by the CFL stability condition of the acoustic term.Numerical results show that the third-order semi-implicit WCNS scheme has higher time accuracy and strong shock wave capture ability.