A general strategy for the optimization of Runge–Kutta schemes for wave propagation phenomena

We analyze optimized explicit Runge–Kutta schemes (RK) for computational aeroacoustics, and wave propagation phenomena in general. Exploiting the analysis developed in [S. Pirozzoli, Performance analysis and optimization of finite-difference schemes for wave propagation problems, J. Comput. Phys. 222 (2007) 809–831], we rigorously evaluate the performance of several time integration schemes in terms of appropriate error and cost metrics, and provide a general strategy to design Runge–Kutta methods tailored for specific applications. We present families of optimized second- and third-order Runge–Kutta schemes with up to seven stages, and describe their implementation in the framework of Williamson’s 2N2N-storage formulation [J.H. Williamson, Low-storage Runge–Kutta schemes, J. Comput. Phys. 35 (1980) 48–56]. Numerical simulations of the 1D linear advection equation and of the 2D linearized Euler equations are performed to demonstrate the validity of the theory and to quantify the improvement provided by optimized schemes.