Simulation of shock-containing flows using a compact finite difference scheme and a weighed essentially non-oscillatory filter

In the present paper, a hybrid filter is introduced to simultaneously preserve the stability and accuracy and also to eliminate unwanted oscillations in the numerical simulation of shock-containing flows. The fourth-order compact finite difference scheme is used for the spatial discretization and the third-order Runge-Kutta scheme is used for the time integration. After each time-step, the hybrid filter is applied on the results. The filter is composed of a linear sixth-order filter and the dissipative part of the fifth-order weighted essentially non-oscillatory scheme. Using a shock-detecting sensor, the hybrid filter reduces to the linear sixth-order filter in smooth regions and to the fifth-order weighted essentially non-oscillatory filter in shock regions in order to eliminate unwanted oscillations produced by the non-dissipative spatial discretization method. The filter performance and accuracy of the results are examined through several test cases including the linear wave equation and one- and two-dimensional Euler equations of gas dynamics. The results are compared by that of a hybrid filter which is composed of the linear sixth-order and the second-order linear filter and that of the fifth-order weighted essentially non-oscillatory scheme.