Nonuniform time-step Runge-Kutta discontinuous Galerkin method for Computational Aeroacoustics

In computational aeroacoustics (CAA) simulations, discontinuous Galerkin space discretization (DG) in conjunction with Runge-Kutta time integration (RK), which is so called Runge-Kutta discontinuous Galerkin method (RKDG), has been an attractive alternative to the finite difference based high-order numerical approaches. However, when it comes to complex physical problems, especially the ones involving irregular geometries, an expensive computational cost are usually required for time-accurate solution, because the time step size of an explicit RK scheme is limited by the smallest grid size in the computational domain. For computational efficiency, high-order RK method with nonuniform time step sizes on nonuniform meshes is developed in this paper. On the elements neighboring the interfaces of grids with different time step sizes, the values at intermediate stages of RK time integration are coupled suitably to realize the stable communication of solutions at those interfaces with minimal dissipation and dispersion errors. A linear coupling procedure is described based upon the general form of a p-stage RK scheme, and also extended to the high-order RK schemes frequently used in simulation of fluid flow and acoustics, including the third order TVD scheme, and low-storage low dissipation and low dispersion schemes. In addition, an eigenvalue analysis of nonuniform time-step RK integration combined with DG method on a nonuniform grid is carried out for the stability property. For verification, numerical experiments on one-dimensional and two-dimensional linear problems are conducted to illustrate the stability and accuracy of proposed nonuniform time-step RKDG scheme. Application to a one-dimensional nonlinear problem is also investigated.