A general strategy for the optimization of Runge-Kutta schemes for wave propagation phenomena. (English) Zbl 1273.76294
Summary: We analyze optimized explicit Runge-Kutta schemes (RK) for computational aeroacoustics, and wave propagation phenomena in general. Exploiting the analysis developed in [S. Pirozzoli, J. Comput. Phys. 222, No. 2, 809–831 (2007; Zbl 1158.76382)], 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 \(2N\)-storage formulation [J.H. Williamson, J. Comput. Phys. 35, 48–56 (1980; Zbl 0425.65038)]. 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.
MSC:
| 76M20 | Finite difference methods applied to problems in fluid mechanics |
| 65L05 | Numerical methods for initial value problems involving ordinary differential equations |
Keywords:
computational aeroacoustics; Runge-Kutta schemes; optimized algorithms; numerical dispersion; numerical dissipationReferences:
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