On implicit time methods and discontinuous Galerkin space reconstruction for conservation laws
<|reference_start|>On implicit time methods and discontinuous Galerkin space reconstruction for conservation laws: In many physical scenarios governed by systems of hyperbolic conservation laws, stiffness necessitates small time-steps due to the stringent CFL stability criterion. Implicit time integration schemes leverage superior stability properties enabling the selection of time-steps based solely on accuracy requirements, thereby bypassing the need for minute time-steps. In this work, we consider high-order diagonally implicit Runge-Kutta time integration methods coupled with discontinuous Galerkin space reconstructions for stiff hyperbolic systems. We analyze dispersion-diffusion properties to select the best combination of the space-time discretization for high Courant numbers, in terms of controlling spurious oscillations. Working with high-order methods, however, requires to introduce local space limiters which make the whole implicit scheme highly nonlinear. Therefore, we propose to use appropriate space limiters that can be precomputed on a first-order predictor of the solution. This approach follows the methodology proposed by Puppo et al. (Commun. Comput. Phys., 2024) for high-order finite volume schemes. Numerical experiments involve both scalar equations and systems.<|reference_end|>
@article{briani2024on,
    title={On implicit time methods and discontinuous Galerkin space reconstruction
  for conservation laws},
    author={Maya Briani, Gabriella Puppo, Giuseppe Visconti},
    journal={arXiv preprint arXiv:2410.05901},
    year={2024},
    number={Roma01.Math.NA},
    archivePrefix={arXiv},
    eprint={2410.05901},
    primaryClass={math.NA cs.NA}
}