Nonuniform-time-step explicit runge-kutta scheme for high-order finite difference method

Explicit Runge-Kutta method is one of the most popular approaches for time-accurate simulation in aeroacoustics and aerodynamics field, because of its stronger stability property and easier implementation comparing with other kinds of explicit method. However, when dealing with the problems involving irregular geometries or multi-scale phenomena where the grid should be refined in specific region and could be coarsen in other area, the time step size uniform in the entire computational domain is limited by the smallest grid size based on the stability restriction, and that would result in excessive computational assumption. Local time step strategy that adopts local time step sizes satisfying the local stability requirement in different grid block is an effective way to reduce the overall computational cost for time integration. In present paper, the nonuniform-time-step explicit Runge-Kutta scheme, which makes the local time-step strategy applicable for explicit Runge-Kutta family without any interpolation in time and has been introduced in the framework of discontinuous Galerkin method, is further developed for high order finite difference methods that are used widely and reliably in various aeroacoustic and aerodynamic problems. The stability of the scheme in combination with the typical high order difference scheme, e.g. dispersion-relation-preserving scheme, is studied. Then the numerical tests are carried out in 1D and 2D acoustic problems for validation, and discussed are also the computational cost reduction caused by the developed algorithm.