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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MDPI AG
2021-10-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/21/2718 |
| _version_ | 1797512098604908544 |
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| author | Lina Zhang Xin Wu Enwei Liang |
| author_facet | Lina Zhang Xin Wu Enwei Liang |
| author_sort | Lina Zhang |
| collection | DOAJ |
| 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. |
| first_indexed | 2024-03-10T05:57:06Z |
| format | Article |
| id | doaj.art-b22e3e6ed7754ffa8c8a352450b3c3d6 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T05:57:06Z |
| publishDate | 2021-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-b22e3e6ed7754ffa8c8a352450b3c3d62023-11-22T21:17:52ZengMDPI AGMathematics2227-73902021-10-01921271810.3390/math9212718Adjustment of Force–Gradient Operator in Symplectic MethodsLina Zhang0Xin Wu1Enwei Liang2School of Physical Science and Technology, Guangxi University, Nanning 530004, ChinaSchool of Physical Science and Technology, Guangxi University, Nanning 530004, ChinaSchool of Physical Science and Technology, Guangxi University, Nanning 530004, ChinaMany 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.https://www.mdpi.com/2227-7390/9/21/2718symplectic integrationforce gradientchaosHamiltonian systems |
| spellingShingle | Lina Zhang Xin Wu Enwei Liang Adjustment of Force–Gradient Operator in Symplectic Methods Mathematics symplectic integration force gradient chaos Hamiltonian systems |
| title | 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_short | Adjustment of Force–Gradient Operator in Symplectic Methods |
| title_sort | adjustment of force gradient operator in symplectic methods |
| topic | symplectic integration force gradient chaos Hamiltonian systems |
| url | https://www.mdpi.com/2227-7390/9/21/2718 |
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