An Efficient and Accurate Adaptive Time-Stepping Method for the Landau–Lifshitz Equation
This article presents an efficient and accurate adaptive time-stepping finite difference method (FDM) for solving the Landau–Lifshitz (LL) equation, which is an important mathematical model in understanding magnetic materials and processes. Our proposed algorithm strategically selects an adaptive ti...
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| author | Hyundong Kim Soobin Kwak Moumni Mohammed Seungyoon Kang Seokjun Ham Junseok Kim |
| author_facet | Hyundong Kim Soobin Kwak Moumni Mohammed Seungyoon Kang Seokjun Ham Junseok Kim |
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| description | This article presents an efficient and accurate adaptive time-stepping finite difference method (FDM) for solving the Landau–Lifshitz (LL) equation, which is an important mathematical model in understanding magnetic materials and processes. Our proposed algorithm strategically selects an adaptive time step, ensuring that the maximum displacement falls within a predefined tolerance threshold. Furthermore, this adaptive approach allows the utilization of larger time steps near equilibrium states and results in faster computations. For example, we introduce a numerical test where the adaptive time step decreases from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>3.05</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>7</mn></mrow></msup></mrow></semantics></math></inline-formula> to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>3.52</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>9</mn></mrow></msup></mrow></semantics></math></inline-formula>. If a uniform time step is applied, around a 100 times smaller time step must be applied at unnecessary cases. To demonstrate the high performance of our proposed algorithm, we conduct several characteristic benchmark tests. The computational results confirm that the proposed algorithm is efficient and accurate. Overall, our adaptive time-stepping FDM offers a promising solution for accurately and efficiently solving the LL equation and contributes to advancements in the understanding and analysis of magnetic phenomena. |
| format | Article |
| id | doaj-art-ee0c6d72f5804b74a40fc3c94c1869a9 |
| institution | Kabale University |
| issn | 1999-4893 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | MDPI AG |
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| series | Algorithms |
| spelling | doaj-art-ee0c6d72f5804b74a40fc3c94c1869a92025-01-24T13:17:25ZengMDPI AGAlgorithms1999-48932024-12-01181110.3390/a18010001An Efficient and Accurate Adaptive Time-Stepping Method for the Landau–Lifshitz EquationHyundong Kim0Soobin Kwak1Moumni Mohammed2Seungyoon Kang3Seokjun Ham4Junseok Kim5Department of Mathematics and Physics, Gangneung-Wonju National University, Gangneung 25457, Republic of KoreaDepartment of Mathematics, Korea University, Seoul 02841, Republic of KoreaMAMCS Group, FST Errachidia, Moulay Ismail University of Meknes, Boutalamine, P.O. Box 509, 52000 Errachidia, MoroccoDepartment of Mathematics, Korea University, Seoul 02841, Republic of KoreaDepartment of Mathematics, Korea University, Seoul 02841, Republic of KoreaDepartment of Mathematics, Korea University, Seoul 02841, Republic of KoreaThis article presents an efficient and accurate adaptive time-stepping finite difference method (FDM) for solving the Landau–Lifshitz (LL) equation, which is an important mathematical model in understanding magnetic materials and processes. Our proposed algorithm strategically selects an adaptive time step, ensuring that the maximum displacement falls within a predefined tolerance threshold. Furthermore, this adaptive approach allows the utilization of larger time steps near equilibrium states and results in faster computations. For example, we introduce a numerical test where the adaptive time step decreases from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>3.05</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>7</mn></mrow></msup></mrow></semantics></math></inline-formula> to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>3.52</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>9</mn></mrow></msup></mrow></semantics></math></inline-formula>. If a uniform time step is applied, around a 100 times smaller time step must be applied at unnecessary cases. To demonstrate the high performance of our proposed algorithm, we conduct several characteristic benchmark tests. The computational results confirm that the proposed algorithm is efficient and accurate. Overall, our adaptive time-stepping FDM offers a promising solution for accurately and efficiently solving the LL equation and contributes to advancements in the understanding and analysis of magnetic phenomena.https://www.mdpi.com/1999-4893/18/1/1adaptive time-stepping algorithmLandau–Lifshitz equationfinite difference method |
| spellingShingle | Hyundong Kim Soobin Kwak Moumni Mohammed Seungyoon Kang Seokjun Ham Junseok Kim An Efficient and Accurate Adaptive Time-Stepping Method for the Landau–Lifshitz Equation Algorithms adaptive time-stepping algorithm Landau–Lifshitz equation finite difference method |
| title | An Efficient and Accurate Adaptive Time-Stepping Method for the Landau–Lifshitz Equation |
| title_full | An Efficient and Accurate Adaptive Time-Stepping Method for the Landau–Lifshitz Equation |
| title_fullStr | An Efficient and Accurate Adaptive Time-Stepping Method for the Landau–Lifshitz Equation |
| title_full_unstemmed | An Efficient and Accurate Adaptive Time-Stepping Method for the Landau–Lifshitz Equation |
| title_short | An Efficient and Accurate Adaptive Time-Stepping Method for the Landau–Lifshitz Equation |
| title_sort | efficient and accurate adaptive time stepping method for the landau lifshitz equation |
| topic | adaptive time-stepping algorithm Landau–Lifshitz equation finite difference method |
| url | https://www.mdpi.com/1999-4893/18/1/1 |
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