L. Chacon, Jason Hamilton, Natalia Krasheninnikova
{"title":"磁化等离子体中强各向异性输运方程的稳健四阶有限差分离散法","authors":"L. Chacon, Jason Hamilton, Natalia Krasheninnikova","doi":"arxiv-2409.06070","DOIUrl":null,"url":null,"abstract":"We propose a second-order temporally implicit, fourth-order-accurate spatial\ndiscretization scheme for the strongly anisotropic heat transport equation\ncharacteristic of hot, fusion-grade plasmas. Following [Du Toit et al., Comp.\nPhys. Comm., 228 (2018)], the scheme transforms mixed-derivative diffusion\nfluxes (which are responsible for the lack of a discrete maximum principle)\ninto nonlinear advective fluxes, amenable to nonlinear-solver-friendly\nmonotonicity-preserving limiters. The scheme enables accurate multi-dimensional\nheat transport simulations with up to seven orders of magnitude of\nheat-transport-coefficient anisotropies with low cross-field numerical error\npollution and excellent algorithmic performance, with the number of linear\niterations scaling very weakly with grid resolution and grid anisotropy, and\nscaling with the square-root of the implicit timestep. We propose a multigrid\npreconditioning strategy based on a second-order-accurate approximation that\nrenders the scheme efficient and scalable under grid refinement. Several\nnumerical tests are presented that display the expected spatial convergence\nrates and strong algorithmic performance, including fully nonlinear\nmagnetohydrodynamics simulations of kink instabilities in a Bennett pinch in 2D\nhelical geometry and of ITER in 3D toroidal geometry.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"98 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A robust fourth-order finite-difference discretization for the strongly anisotropic transport equation in magnetized plasmas\",\"authors\":\"L. Chacon, Jason Hamilton, Natalia Krasheninnikova\",\"doi\":\"arxiv-2409.06070\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We propose a second-order temporally implicit, fourth-order-accurate spatial\\ndiscretization scheme for the strongly anisotropic heat transport equation\\ncharacteristic of hot, fusion-grade plasmas. Following [Du Toit et al., Comp.\\nPhys. Comm., 228 (2018)], the scheme transforms mixed-derivative diffusion\\nfluxes (which are responsible for the lack of a discrete maximum principle)\\ninto nonlinear advective fluxes, amenable to nonlinear-solver-friendly\\nmonotonicity-preserving limiters. The scheme enables accurate multi-dimensional\\nheat transport simulations with up to seven orders of magnitude of\\nheat-transport-coefficient anisotropies with low cross-field numerical error\\npollution and excellent algorithmic performance, with the number of linear\\niterations scaling very weakly with grid resolution and grid anisotropy, and\\nscaling with the square-root of the implicit timestep. We propose a multigrid\\npreconditioning strategy based on a second-order-accurate approximation that\\nrenders the scheme efficient and scalable under grid refinement. Several\\nnumerical tests are presented that display the expected spatial convergence\\nrates and strong algorithmic performance, including fully nonlinear\\nmagnetohydrodynamics simulations of kink instabilities in a Bennett pinch in 2D\\nhelical geometry and of ITER in 3D toroidal geometry.\",\"PeriodicalId\":501162,\"journal\":{\"name\":\"arXiv - MATH - Numerical Analysis\",\"volume\":\"98 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-09\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Numerical Analysis\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.06070\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Numerical Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.06070","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
A robust fourth-order finite-difference discretization for the strongly anisotropic transport equation in magnetized plasmas
We propose a second-order temporally implicit, fourth-order-accurate spatial
discretization scheme for the strongly anisotropic heat transport equation
characteristic of hot, fusion-grade plasmas. Following [Du Toit et al., Comp.
Phys. Comm., 228 (2018)], the scheme transforms mixed-derivative diffusion
fluxes (which are responsible for the lack of a discrete maximum principle)
into nonlinear advective fluxes, amenable to nonlinear-solver-friendly
monotonicity-preserving limiters. The scheme enables accurate multi-dimensional
heat transport simulations with up to seven orders of magnitude of
heat-transport-coefficient anisotropies with low cross-field numerical error
pollution and excellent algorithmic performance, with the number of linear
iterations scaling very weakly with grid resolution and grid anisotropy, and
scaling with the square-root of the implicit timestep. We propose a multigrid
preconditioning strategy based on a second-order-accurate approximation that
renders the scheme efficient and scalable under grid refinement. Several
numerical tests are presented that display the expected spatial convergence
rates and strong algorithmic performance, including fully nonlinear
magnetohydrodynamics simulations of kink instabilities in a Bennett pinch in 2D
helical geometry and of ITER in 3D toroidal geometry.