Krylov-based Adaptive-Rank Implicit Time Integrators for Stiff Problems with Application to Nonlinear Fokker-Planck Kinetic Models
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866916192362430464 |
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| author | Kahza, Hamad El Taitano, William Qiu, Jing-Mei Chacón, Luis |
| author_facet | Kahza, Hamad El Taitano, William Qiu, Jing-Mei Chacón, Luis |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_03119 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Krylov-based Adaptive-Rank Implicit Time Integrators for Stiff Problems with Application to Nonlinear Fokker-Planck Kinetic Models Kahza, Hamad El Taitano, William Qiu, Jing-Mei Chacón, Luis Numerical Analysis Mathematical Physics 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. |
| title | Krylov-based Adaptive-Rank Implicit Time Integrators for Stiff Problems with Application to Nonlinear Fokker-Planck Kinetic Models |
| topic | Numerical Analysis Mathematical Physics |
| url | https://arxiv.org/abs/2404.03119 |