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Multirate infinitesimal step methods for atmospheric flow simulation

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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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Authors and Affiliations

  1. Fachrichtung Mathematik, Inst. f. wiss. Rechnen, TU Dresden, 01062, Dresden, Germany

    Jörg Wensch & Alexander Galant

  2. Inst. f. Troposphärenforschung, Permoserstrasse 15, 04318, Leipzig, Germany

    Oswald Knoth

Authors
  1. Jörg Wensch
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  2. Oswald Knoth
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  3. Alexander Galant
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Correspondence to Jörg Wensch.

Additional information

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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