High order stable multi-domain hybrid RKDG and WENO-FD methods^*

Recently, a kind of high order hybrid methods based on Runge-Kutta discontinuous Galerkin (RKDG) method and weighted essentially non-oscillatory finite difference (WENO-FD) scheme was proposed. Those methods are computationally efficient, however stable problems might sometimes be encountered in practical applications. In this work, we first analyze the linear stabilities of those methods based on the Heuristic theory. We find that the conservative hybrid method is linearly unstable if the numerical flux at the coupling interface is chosen to be ‘downstream’. Then we introduce two ways of healing this defect. One is to choose the numerical flux at the coupling interface to be ‘upstream’. The other is to employ a slope limiter function to enforce the hybrid method satisfying the local total variation diminishing (TVD) condition. In the end, numerical experiments are provided to validate the effectiveness of the proposed methods.