Stability Analysis on the Hypersonic Numerical Shocks for Finite-volume WENO Schemes
High-order schemes offer significant advantages in improving computational accuracy and capturing flow details.However,when simulating multidimensional strong shocks,high-order schemes often suffer shock instability problems,which limit their applications in supersonic/hypersonic flow simulations.In the current study,the matrix stability analysis method is established to quantitatively analyze the stability of the third-order WENO finite-volume scheme in capturing strong shocks.And by using the matrix stability analysis method,the shock instability problem of the third-order scheme is studied.Results indicate that compared with first-order schemes,the third-order scheme has more severe shock instability problems.Under spatial third-order accuracy,the properties of Riemann solvers significantly affect the shock instability problem.Low-dissipative solvers,such as Roe,often lead to shock instability,while the third-order scheme can capture strong shocks when high-dissipation solvers are employed.Furthermore,it has been demonstrated that the intensity of the shock is positively correlated with the shock instability problem,but its impact weakens or even disappears beyond a certain level.Additionally,the spatial location of numerical shock instability is investigated in the current work,and it is found that for the third-order scheme,the shock instability originates from the numerical shock structure.As a result,the intermediate states inside the shock structure are crucial factors affecting the stability of the scheme in capturing strong shocks.When the intermediate states inside the shock structure are closer to the downstream states,the computation will be more stable.This paper provides an effective method for analyzing the shock instability problem of third-order WENO finite-volume schemes and investigates the underlying mechanisms of shock instability problem for the third-order scheme,laying a foundation for further understanding the inherent mechanisms of shock instability problems in high-order schemes and developing stable high-order schemes.
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