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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| Main Authors: | , , , , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2024-12-01
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| Series: | Algorithms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/1999-4893/18/1/1 |
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| Summary: | 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. |
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| ISSN: | 1999-4893 |