A Local Macroscopic Conservative (LoMaC) Low Rank Tensor Method for the Vlasov Dynamics
In this paper, we propose a novel Local Macroscopic Conservative (LoMaC) low rank tensor method for simulating the Vlasov-Poisson (VP) system. The LoMaC property refers to the exact local conservation of macroscopic mass, momentum and energy at the discrete level. This is a follow-up work of our pre...
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| Published in | Journal of scientific computing Vol. 101; no. 3; p. 61 |
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| Abstract | In this paper, we propose a novel Local Macroscopic Conservative (LoMaC) low rank tensor method for simulating the Vlasov-Poisson (VP) system. The LoMaC property refers to the exact local conservation of macroscopic mass, momentum and energy at the discrete level. This is a follow-up work of our previous development of a conservative low rank tensor approach for Vlasov dynamics (
arXiv:2201.10397
). In that work, we applied a low rank tensor method with a conservative singular value decomposition to the high dimensional VP system to mitigate the curse of dimensionality, while maintaining the local conservation of mass and momentum. However, energy conservation is not guaranteed, which is a critical property to avoid unphysical plasma self-heating or cooling. The new ingredient in the LoMaC low rank tensor algorithm is that we simultaneously evolve the macroscopic conservation laws of mass, momentum and energy using a flux-difference form with kinetic flux vector splitting; then the LoMaC property is realized by projecting the low rank kinetic solution onto a subspace that shares the same macroscopic observables by a conservative orthogonal projection. The algorithm is extended to the high dimensional problems by hierarchical Tuck decomposition of solution tensors and a corresponding conservative projection algorithm. Extensive numerical tests on the VP system are showcased for the algorithm’s efficacy. |
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| AbstractList | In this paper, we propose a novel Local Macroscopic Conservative (LoMaC) low rank tensor method for simulating the Vlasov-Poisson (VP) system. The LoMaC property refers to the exact local conservation of macroscopic mass, momentum and energy at the discrete level. This is a follow-up work of our previous development of a conservative low rank tensor approach for Vlasov dynamics (
arXiv:2201.10397
). In that work, we applied a low rank tensor method with a conservative singular value decomposition to the high dimensional VP system to mitigate the curse of dimensionality, while maintaining the local conservation of mass and momentum. However, energy conservation is not guaranteed, which is a critical property to avoid unphysical plasma self-heating or cooling. The new ingredient in the LoMaC low rank tensor algorithm is that we simultaneously evolve the macroscopic conservation laws of mass, momentum and energy using a flux-difference form with kinetic flux vector splitting; then the LoMaC property is realized by projecting the low rank kinetic solution onto a subspace that shares the same macroscopic observables by a conservative orthogonal projection. The algorithm is extended to the high dimensional problems by hierarchical Tuck decomposition of solution tensors and a corresponding conservative projection algorithm. Extensive numerical tests on the VP system are showcased for the algorithm’s efficacy. Abstract In this paper, we propose a novel Local Macroscopic Conservative (LoMaC) low rank tensor method for simulating the Vlasov-Poisson (VP) system. The LoMaC property refers to the exact local conservation of macroscopic mass, momentum and energy at the discrete level. This is a follow-up work of our previous development of a conservative low rank tensor approach for Vlasov dynamics ( <ext-link ext-link-type='uri' href='http://arxiv.org/abs/2201.10397'>arXiv:2201.10397</ext-link> ). In that work, we applied a low rank tensor method with a conservative singular value decomposition to the high dimensional VP system to mitigate the curse of dimensionality, while maintaining the local conservation of mass and momentum. However, energy conservation is not guaranteed, which is a critical property to avoid unphysical plasma self-heating or cooling. The new ingredient in the LoMaC low rank tensor algorithm is that we simultaneously evolve the macroscopic conservation laws of mass, momentum and energy using a flux-difference form with kinetic flux vector splitting; then the LoMaC property is realized by projecting the low rank kinetic solution onto a subspace that shares the same macroscopic observables by a conservative orthogonal projection. The algorithm is extended to the high dimensional problems by hierarchical Tuck decomposition of solution tensors and a corresponding conservative projection algorithm. Extensive numerical tests on the VP system are showcased for the algorithm’s efficacy. In this paper, we propose a novel Local Macroscopic Conservative (LoMaC) low rank tensor method for simulating the Vlasov-Poisson (VP) system. The LoMaC property refers to the exact local conservation of macroscopic mass, momentum and energy at the discrete level. This is a follow-up work of our previous development of a conservative low rank tensor approach for Vlasov dynamics (arXiv:2201.10397). In that work, we applied a low rank tensor method with a conservative singular value decomposition to the high dimensional VP system to mitigate the curse of dimensionality, while maintaining the local conservation of mass and momentum. However, energy conservation is not guaranteed, which is a critical property to avoid unphysical plasma self-heating or cooling. The new ingredient in the LoMaC low rank tensor algorithm is that we simultaneously evolve the macroscopic conservation laws of mass, momentum and energy using a flux-difference form with kinetic flux vector splitting; then the LoMaC property is realized by projecting the low rank kinetic solution onto a subspace that shares the same macroscopic observables by a conservative orthogonal projection. The algorithm is extended to the high dimensional problems by hierarchical Tuck decomposition of solution tensors and a corresponding conservative projection algorithm. Extensive numerical tests on the VP system are showcased for the algorithm’s efficacy. |
| ArticleNumber | 61 |
| Author | Guo, Wei Qiu, Jing-Mei |
| Author_xml | – sequence: 1 givenname: Wei surname: Guo fullname: Guo, Wei organization: Department of Mathematics and Statistics, Texas Tech University – sequence: 2 givenname: Jing-Mei orcidid: 0000-0002-3462-188X surname: Qiu fullname: Qiu, Jing-Mei email: jingqiu@udel.edu organization: Department of Mathematical Sciences, University of Delaware |
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| Keywords | Hierarchical Tucker decomposition of tensors Vlasov Dynamics Energy conservation Low rank Conservative SVD LoMaC |
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| References_xml | – reference: ChengYChristliebAJZhongXEnergy-conserving discontinuous Galerkin methods for the Vlasov–Ampere systemJ. Comput. Phys.2014256630655311742710.1016/j.jcp.2013.09.013 – reference: OseledetsIVTyrtyshnikovEEBreaking the curse of dimensionality, or how to use SVD in many dimensionsSIAM J. Sci. Comput.200931537443759255656010.1137/090748330 – reference: Guo, W., Ema, J. F., Qiu, J.-M.: A local macroscopic conservative (LoMac) low rank tensor method with the discontinuous Galerkin method for the Vlasov dynamics (2022). arXiv preprint arXiv:2210.07208 – reference: De LathauwerLDe MoorBVandewalleJA multilinear singular value decompositionSIAM J. Matrix Anal. Appl.200021412531278178027210.1137/S0895479896305696 – reference: GrasedyckLHierarchical singular value decomposition of tensorsSIAM J. Matrix Anal. Appl.201031420292054267895510.1137/090764189 – reference: DawsonJParticle simulation of plasmasRev. Mod. Phys.198355240310.1103/RevModPhys.55.403 – reference: TaoZGuoWChengYSparse grid discontinuous Galerkin methods for the Vlasov–Maxwell systemJ. Comput. Phys. X201931000224116077 – reference: Allmann-Rahn, F., Grauer, R., Kormann, K.: A parallel low-rank solver for the six-dimensional Vlasov–Maxwell equations (2022). arXiv preprint arXiv:2201.03471 – reference: Einkemmer, L., Kusch, J., Schotthöfer, S.: Conservation properties of the augmented basis update & Galerkin integrator for kinetic problems (2023). arXiv preprint arXiv:2311.06399 – reference: KoldaTGBaderBWTensor decompositions and applicationsSIAM Rev.2009513455500253505610.1137/07070111X – reference: de Dios, B. A., Hajian, S.: High order and energy preserving discontinuous Galerkin methods for the Vlasov–Poisson system (2012). arXiv preprint arXiv:1209.4025 – reference: MandalJDeshpandeSKinetic flux vector splitting for Euler equationsComput. 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| Snippet | In this paper, we propose a novel Local Macroscopic Conservative (LoMaC) low rank tensor method for simulating the Vlasov-Poisson (VP) system. The LoMaC... Abstract In this paper, we propose a novel Local Macroscopic Conservative (LoMaC) low rank tensor method for simulating the Vlasov-Poisson (VP) system. The... |
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| SubjectTerms | Algorithms Approximation Computational Mathematics and Numerical Analysis Conservation Energy conservation Flux vector splitting Mathematical and Computational Engineering Mathematical and Computational Physics Mathematics Mathematics and Statistics Momentum Simulation Singular value decomposition Tensors Theoretical |
| Title | A Local Macroscopic Conservative (LoMaC) Low Rank Tensor Method for the Vlasov Dynamics |
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