Krylov-based Adaptive-Rank Implicit Time Integrators for Stiff Problems with Application to Nonlinear Fokker-Planck Kinetic Models
<|reference_start|>Krylov-based Adaptive-Rank Implicit Time Integrators for Stiff Problems with Application to Nonlinear Fokker-Planck Kinetic Models: We propose a high order adaptive-rank implicit integrators for stiff time-dependent PDEs, leveraging extended Krylov subspaces to efficiently and adaptively populate low-rank solution bases. This allows for the accurate representation of solutions with significantly reduced computational costs. We further introduce an efficient mechanism for residual evaluation and an adaptive rank-seeking strategy that optimizes low-rank settings based on a comparison between the residual size and the local truncation errors of the time-stepping discretization. We demonstrate our approach with the challenging Lenard-Bernstein Fokker-Planck (LBFP) nonlinear equation, which describes collisional processes in a fully ionized plasma. The preservation of {the equilibrium state} is achieved through the Chang-Cooper discretization, and strict conservation of mass, momentum and energy via a Locally Macroscopic Conservative (LoMaC) procedure. The development of implicit adaptive-rank integrators, demonstrated here up to third-order temporal accuracy via diagonally implicit Runge-Kutta schemes, showcases superior performance in terms of accuracy, computational efficiency, equilibrium preservation, and conservation of macroscopic moments. This study offers a starting point for developing scalable, efficient, and accurate methods for high-dimensional time-dependent problems.<|reference_end|>
@article{kahza2024krylov-based,
    title={Krylov-based Adaptive-Rank Implicit Time Integrators for Stiff Problems
  with Application to Nonlinear Fokker-Planck Kinetic Models},
    author={Hamad El Kahza, William Taitano, Jing-Mei Qiu, Luis Chac'on},
    journal={arXiv preprint arXiv:2404.03119},
    year={2024},
    archivePrefix={arXiv},
    eprint={2404.03119},
    primaryClass={math.NA cs.NA math-ph math.MP}
}