Numerical Researches Of Hyperbolic Problems Based On Stochastic Galerkin Method And High Accuracy Scheme

Stochastic factors often exist in realistic fluid flow problems,and in numerical simulation,uncertainties may implicit in parameters or physical quantities in fluid flow model.With the development of science and technology,solving the random fluid dynamics problems becomes possible.Stochastic fluid dynamics problems are hot and difficult problems in computational fluid dynamics.In solving stochastic hyperbolic conservation laws,the numerical methods currently used often fail to guarantee the diagonalization of Jacobian matrix,resulting in the loss of hyperbolicity of the transformed problems.On the other hand,accuracy of inviscid flux has always been a hot topic in computational fluid dynamics.With focusing on the above issues,the main research contents in this paper are as follows.(1).The Galerkin projection on stochastic hyperbolic problem has been investigated.Based on Legendre orthogonal polynomial basis function,a onedimensional stochastic hyperbolic model is transformed into a deterministic hyperbolic conservation law by using stochastic Galerkin method.In order to guarantee the hyperbolicity of the system,the approximation Galerkin Jacobian matrix is introduced.(2).Six flux schemes are numerically validated.Roe Riemann solver with high robustness is selected to compute hyperbolic conservation flux,and HartenHyman entropy corrector is carried out.(3).The left and right state values in Roe flux are reconstructed by using the fifth-order WENO-Z scheme.The classical one-dimensional and twodimensional deterministic Euler equations are computed and compared with fifth-order WENO-JS scheme.The results imply that fifth-order Roe-WENO-Z scheme has not only higher calculation accuracy,but also higher resolution for shock wave and vortices.(4).The feasibility of using polynomial chaos theory to deal with random variables is verified.Based on stochastic Galerkin method and fifth order RoeWENO-Z scheme,the stochastic Burgers and one-dimensional Euler equations are computed,which lays a gorgeous foundation for the solution of twodimensional stochastic flow.