High Order Accurate Difference Methods For Hyperbolic Conservation Law Equations In Fluid Dynamics

Hyperbolic conservation law equations are very important partial differential equations in fluid dynamics,the characteristic of its solution is that no matter how smooth the initial and boundary values are,the solution is likely to be discontinous as time goes on.Therefore,it is a difficult tast to obtain the solution of the equations.In recent years,high-order accuracy and high resolution numerical methods of hyperbolic conservation law equations have been developed rapidly,the weighted essentially non-oscillatory(WEND)method is one of the most efficient method that has been developed in the last two decades,with the greatest advantage being that it is high precision and easy to implement,however,the accuracy of classical WENO schemes can decrease near critical points of smooth function,the resolution is also not enough for strong discontinuity.In order to solve this problem,based on WENO difference method,several high-order accuracy,high resolution and low dissipaton WENO difference schemes are obtained by improving the local and global smoothness indicators and combining with nonlinear WENO interpolation and high-order compact difference scheme.Finally,combined with the well-balanced method of shallow water equation with source term,the numerical simulation of dam-break flow and other hydrodynamic problems is carried out.The main contents and achievements are as follows:.1.An improved third-order accuracy WENO difference schemeUnder the basic framework of tratidtional WENO-Z scheme,a new global smoothness indicater that contains parameter is obtained by introducing parameter P and expanding the smootheness indicators of third-order WENO scheme with Taylor expansion.The optimal value of parameter P is got under the condition of reache third-order convergence accuracy,an improved third-order WENO difference scheme(M-WENO3-1)is obtained;Another new improved third-order WENO difference scheme(M-WENO3-2)is obtained by reselecting the computational stencils of the third-order WENO difference scheme,combining the local smoothness indicator of each stencil with weighted linear combination,constructing a new global smoothness indicator,establishing a new nonlinear weight and introducing adjustable linear weight and large-template reconstruction cell boundary numerical flux.Finally,the convergence of these two schemes are proved,numerical experiments are carried to verify the accuracy of the schemes and the resolution for discontinuity problem.2.An improved fifth-order accuracy WENO difference schemeA new fifth-order WENO difference scheme is obtained by reselecting the computational stencils of the third-order WENO difference scheme,combining the local smoothness indicator of each stencil with weighted linear combination,constructing a new global smoothness indicator and establishing a new nonlinear weight,the cell boundary numerical flux is computed by combianed four reconstructed polynomials on large template and two secondary reconstruction polynomials on two small templates.The convergence of the scheme is also proved,numerical experiments are carried to verify the accuracy of the scheme and the resolution for discontinuity problem.3.High-order compact nonlinear WENO difference schemeThe fifth-order function value at cell interface can be obtained by combining the nonlinear weights of WENO scheme established in 2 with WENO interpolation.Using the four interpolation polynomial on the large template and the second interpolation polynomials on the two small templates,the first derivative value of grid points are got by the fourth order,six order compact difference scheme,and combined with the boundary conditions that match the interior point precision,the fourth-order and fifith-order compact nonlinear WENO scheme are obtained respectively,they are denoted as MC-WENO4 and MC-WENO5.Numerical experiments are carried to verify the accuracy of the scheme and the resolution for discontinuity problem.4.Numerical simulation of dam-break and other hydraulic phenomena by combining WENO difference scheme with well-balanced method of source term shallow water equationsWENO difference schemes established above are used to simulate the dam-break and other hydraulic phenomena.Firstly,one-and two-dimensional ideal dam break problems are simulated.Then,the dam-break problem with different bottom slope source terms and other perturbation problems are numerically calculated combined with the existing well-balanced method.The results show that the simulation effect of the proposed method is satisfactory and the ability to capture shock waves and disturbances is very strong.