Adjustment of Force–Gradient Operator in Symplectic Methods

Adjustment of Force–Gradient Operator in Symplectic Methods

Many force–gradient explicit symplectic integration algorithms have been designed for the Hamiltonian <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>=</mo><mi>T...

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Autores principales: Lina Zhang, Xin Wu, Enwei Liang
Formato: article
Lenguaje:EN
Publicado: MDPI AG 2021
Materias:
symplectic integration
force gradient
chaos
Hamiltonian systems
Mathematics
QA1-939
Acceso en línea:https://doaj.org/article/b22e3e6ed7754ffa8c8a352450b3c3d6
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id oai:doaj.org-article:b22e3e6ed7754ffa8c8a352450b3c3d6
record_format dspace
institution DOAJ
collection DOAJ
language EN
topic symplectic integration
force gradient
chaos
Hamiltonian systems
Mathematics
QA1-939
spellingShingle symplectic integration
force gradient
chaos
Hamiltonian systems
Mathematics
QA1-939
Lina Zhang
Xin Wu
Enwei Liang
Adjustment of Force–Gradient Operator in Symplectic Methods
description Many force–gradient explicit symplectic integration algorithms have been designed for the Hamiltonian <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>=</mo><mi>T</mi><mo>(</mo><mi mathvariant="bold">p</mi><mo>)</mo><mo>+</mo><mi>V</mi><mo>(</mo><mi mathvariant="bold">q</mi><mo>)</mo></mrow></semantics></math></inline-formula> with kinetic energy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mrow><mo>(</mo><mi mathvariant="bold">p</mi><mo>)</mo></mrow><mo>=</mo><msup><mi mathvariant="bold">p</mi><mn>2</mn></msup><mo>/</mo><mn>2</mn></mrow></semantics></math></inline-formula> in the existing references. When a force–gradient operator is appropriately adjusted as a new operator, it is still suitable for a class of Hamiltonian problems <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>=</mo><mi>K</mi><mo>(</mo><mi mathvariant="bold">p</mi><mo>,</mo><mi mathvariant="bold">q</mi><mo>)</mo><mo>+</mo><mi>V</mi><mo>(</mo><mi mathvariant="bold">q</mi><mo>)</mo></mrow></semantics></math></inline-formula> with integrable part <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>K</mi><mrow><mo>(</mo><mi mathvariant="bold">p</mi><mo>,</mo><mi mathvariant="bold">q</mi><mo>)</mo></mrow><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msubsup><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mi>a</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mi>p</mi><mi>i</mi></msub><msub><mi>p</mi><mi>j</mi></msub><mo>+</mo><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mi>b</mi><mi>i</mi></msub><msub><mi>p</mi><mi>i</mi></msub></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>a</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><msub><mi>a</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mrow><mo>(</mo><mi mathvariant="bold">q</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>b</mi><mi>i</mi></msub><mo>=</mo><msub><mi>b</mi><mi>i</mi></msub><mrow><mo>(</mo><mi mathvariant="bold">q</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> are functions of coordinates <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold">q</mi></semantics></math></inline-formula>. The newly adjusted operator is not a force–gradient operator but is similar to the momentum-version operator associated to the potential <i>V</i>. The newly extended (or adjusted) algorithms are no longer solvers of the original Hamiltonian, but are solvers of slightly modified Hamiltonians. They are explicit symplectic integrators with symmetry or time reversibility. Numerical tests show that the standard symplectic integrators without the new operator are generally poorer than the corresponding extended methods with the new operator in computational accuracies and efficiencies. The optimized methods have better accuracies than the corresponding non-optimized counterparts. Among the tested symplectic methods, the two extended optimized seven-stage fourth-order methods of Omelyan, Mryglod and Folk exhibit the best numerical performance. As a result, one of the two optimized algorithms is used to study the orbital dynamical features of a modified Hénon–Heiles system and a spring pendulum. These extended integrators allow for integrations in Hamiltonian problems, such as the spiral structure in self-consistent models of rotating galaxies and the spiral arms in galaxies.
format article
author Lina Zhang
Xin Wu
Enwei Liang
author_facet Lina Zhang
Xin Wu
Enwei Liang
author_sort Lina Zhang
title Adjustment of Force–Gradient Operator in Symplectic Methods
title_short Adjustment of Force–Gradient Operator in Symplectic Methods
title_full Adjustment of Force–Gradient Operator in Symplectic Methods
title_fullStr Adjustment of Force–Gradient Operator in Symplectic Methods
title_full_unstemmed Adjustment of Force–Gradient Operator in Symplectic Methods
title_sort adjustment of force–gradient operator in symplectic methods
publisher MDPI AG
publishDate 2021
url https://doaj.org/article/b22e3e6ed7754ffa8c8a352450b3c3d6
work_keys_str_mv AT linazhang adjustmentofforcegradientoperatorinsymplecticmethods
AT xinwu adjustmentofforcegradientoperatorinsymplecticmethods
AT enweiliang adjustmentofforcegradientoperatorinsymplecticmethods
_version_ 1718431905853997056
spelling oai:doaj.org-article:b22e3e6ed7754ffa8c8a352450b3c3d62021-11-11T18:16:32ZAdjustment of Force–Gradient Operator in Symplectic Methods10.3390/math92127182227-7390https://doaj.org/article/b22e3e6ed7754ffa8c8a352450b3c3d62021-10-01T00:00:00Zhttps://www.mdpi.com/2227-7390/9/21/2718https://doaj.org/toc/2227-7390Many force–gradient explicit symplectic integration algorithms have been designed for the Hamiltonian <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>=</mo><mi>T</mi><mo>(</mo><mi mathvariant="bold">p</mi><mo>)</mo><mo>+</mo><mi>V</mi><mo>(</mo><mi mathvariant="bold">q</mi><mo>)</mo></mrow></semantics></math></inline-formula> with kinetic energy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mrow><mo>(</mo><mi mathvariant="bold">p</mi><mo>)</mo></mrow><mo>=</mo><msup><mi mathvariant="bold">p</mi><mn>2</mn></msup><mo>/</mo><mn>2</mn></mrow></semantics></math></inline-formula> in the existing references. When a force–gradient operator is appropriately adjusted as a new operator, it is still suitable for a class of Hamiltonian problems <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>=</mo><mi>K</mi><mo>(</mo><mi mathvariant="bold">p</mi><mo>,</mo><mi mathvariant="bold">q</mi><mo>)</mo><mo>+</mo><mi>V</mi><mo>(</mo><mi mathvariant="bold">q</mi><mo>)</mo></mrow></semantics></math></inline-formula> with integrable part <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>K</mi><mrow><mo>(</mo><mi mathvariant="bold">p</mi><mo>,</mo><mi mathvariant="bold">q</mi><mo>)</mo></mrow><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msubsup><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mi>a</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mi>p</mi><mi>i</mi></msub><msub><mi>p</mi><mi>j</mi></msub><mo>+</mo><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mi>b</mi><mi>i</mi></msub><msub><mi>p</mi><mi>i</mi></msub></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>a</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><msub><mi>a</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mrow><mo>(</mo><mi mathvariant="bold">q</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>b</mi><mi>i</mi></msub><mo>=</mo><msub><mi>b</mi><mi>i</mi></msub><mrow><mo>(</mo><mi mathvariant="bold">q</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> are functions of coordinates <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold">q</mi></semantics></math></inline-formula>. The newly adjusted operator is not a force–gradient operator but is similar to the momentum-version operator associated to the potential <i>V</i>. The newly extended (or adjusted) algorithms are no longer solvers of the original Hamiltonian, but are solvers of slightly modified Hamiltonians. They are explicit symplectic integrators with symmetry or time reversibility. Numerical tests show that the standard symplectic integrators without the new operator are generally poorer than the corresponding extended methods with the new operator in computational accuracies and efficiencies. The optimized methods have better accuracies than the corresponding non-optimized counterparts. Among the tested symplectic methods, the two extended optimized seven-stage fourth-order methods of Omelyan, Mryglod and Folk exhibit the best numerical performance. As a result, one of the two optimized algorithms is used to study the orbital dynamical features of a modified Hénon–Heiles system and a spring pendulum. These extended integrators allow for integrations in Hamiltonian problems, such as the spiral structure in self-consistent models of rotating galaxies and the spiral arms in galaxies.Lina ZhangXin WuEnwei LiangMDPI AGarticlesymplectic integrationforce gradientchaosHamiltonian systemsMathematicsQA1-939ENMathematics, Vol 9, Iss 2718, p 2718 (2021)

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