Abstract
The numerical solution of the Euler equations requires the treatment of processes in different temporal scales. Sound waves propagate fast compared to advective processes. Based on a spatial discretisation on staggered grids, a multirate time integration procedure is presented here generalising split-explicit Runge-Kutta methods. The advective terms are integrated by a Runge-Kutta method with a macro stepsize restricted by the CFL number. Sound wave terms are treated by small time steps respecting the CFL restriction dictated by the speed of sound.
Split-explicit Runge-Kutta methods are generalised by the inclusion of fixed tendencies of previous stages. The stability barrier for the acoustics equation is relaxed by a factor of two.
Asymptotic order conditions for the low Mach case are given. The relation to commutator-free exponential integrators is discussed. Stability is analysed for the linear acoustic equation. Numerical tests are executed for the linear acoustics and the nonlinear Euler equations.
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Communicated by Stig Skelboe.
This work was supported under the DFG priority program 1276, Metström: Multiple scales in fluid mechanics and meteorology.
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Wensch, J., Knoth, O. & Galant, A. Multirate infinitesimal step methods for atmospheric flow simulation. Bit Numer Math 49, 449–473 (2009). https://doi.org/10.1007/s10543-009-0222-3
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DOI: https://doi.org/10.1007/s10543-009-0222-3