Multirate Time-Integration based on Dynamic ODE Partitioning through Adaptively Refined Meshes for Compressible Fluid Dynamics
<|reference_start|>Multirate Time-Integration based on Dynamic ODE Partitioning through Adaptively Refined Meshes for Compressible Fluid Dynamics: In this paper, we apply the Paired-Explicit Runge-Kutta (P-ERK) schemes by Vermeire et. al. (2019, 2022) to dynamically partitioned systems arising from adaptive mesh refinement. The P-ERK schemes enable multirate time-integration with no changes in the spatial discretization methodology, making them readily implementable in existing codes that employ a method-of-lines approach.   We show that speedup compared to a range of state of the art Runge-Kutta methods can be realized, despite additional overhead due to the dynamic re-assignment of flagging variables and restricting nonlinear stability properties. The effectiveness of the approach is demonstrated for a range of simulation setups for viscous and inviscid convection-dominated compressible flows for which we provide a reproducibility repository.   In addition, we perform a thorough investigation of the nonlinear stability properties of the Paired-Explicit Runge-Kutta schemes regarding limitations due to the violation of monotonicity properties of the underlying spatial discretization. Furthermore, we present a novel approach for estimating the relevant eigenvalues of large Jacobians required for the optimization of stability polynomials.<|reference_end|>
@article{doehring2024multirate,
    title={Multirate Time-Integration based on Dynamic ODE Partitioning through
  Adaptively Refined Meshes for Compressible Fluid Dynamics},
    author={Daniel Doehring, Michael Schlottke-Lakemper, Gregor J. Gassner, Manuel
  Torrilhon},
    journal={arXiv preprint arXiv:2403.05144},
    year={2024},
    doi={10.1016/j.jcp.2024.113223},
    archivePrefix={arXiv},
    eprint={2403.05144},
    primaryClass={math.NA cs.NA math-ph math.MP}
}