{"title":"针对二维非线性延迟分式方程产生的多奇异性问题的变步 L1 方法与时间双网格算法相结合","authors":"","doi":"10.1016/j.cnsns.2024.108270","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper, we focus on the numerical simulation for two-dimensional nonlinear fractional sub-diffusion equations in the presence of time delay. Firstly, we investigate the existence, uniqueness and regularity of the solution for such problems. The theoretical result implies that the solution at <span><math><msup><mrow><mi>τ</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span> is smoother than that at <span><math><msup><mrow><mn>0</mn></mrow><mrow><mo>+</mo></mrow></msup></math></span>, where <span><math><mi>τ</mi></math></span> is a constant time delay, and this is an improvement for the work (Tan et al., 2022). Secondly, a high-order difference scheme based on <span><math><mrow><mi>L</mi><mn>1</mn></mrow></math></span> method is constructed. For the sake of repairing the convergence order in temporal direction and improving the computational efficiency, an efficient time two-grid algorithm based on nonuniform meshes is first developed. The convergence order of the two-grid scheme reaches <span><math><mrow><mi>O</mi><mrow><mo>(</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>F</mi></mrow><mrow><mo>−</mo><mo>min</mo><mrow><mo>{</mo><mi>r</mi><mi>α</mi><mo>,</mo><mn>2</mn><mo>−</mo><mi>α</mi><mo>}</mo></mrow></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>C</mi></mrow><mrow><mo>−</mo><mo>min</mo><mrow><mo>{</mo><mn>2</mn><mi>r</mi><mi>α</mi><mo>,</mo><mn>4</mn><mo>−</mo><mn>2</mn><mi>α</mi><mo>}</mo></mrow></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>h</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>h</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>)</mo></mrow></mrow></math></span>, where <span><math><msub><mrow><mi>N</mi></mrow><mrow><mi>F</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>N</mi></mrow><mrow><mi>C</mi></mrow></msub></math></span> represent the number of the fine and coarse grids respectively, while <span><math><msub><mrow><mi>h</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>h</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> are the space-step sizes. Furthermore, stability and convergence analysis of the proposed scheme are carefully verified by energy method. Finally, numerical experiments are carried out to show the validity of theoretical statements.</p></div>","PeriodicalId":50658,"journal":{"name":"Communications in Nonlinear Science and Numerical Simulation","volume":null,"pages":null},"PeriodicalIF":3.4000,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Variable-step L1 method combined with time two-grid algorithm for multi-singularity problems arising from two-dimensional nonlinear delay fractional equations\",\"authors\":\"\",\"doi\":\"10.1016/j.cnsns.2024.108270\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>In this paper, we focus on the numerical simulation for two-dimensional nonlinear fractional sub-diffusion equations in the presence of time delay. Firstly, we investigate the existence, uniqueness and regularity of the solution for such problems. The theoretical result implies that the solution at <span><math><msup><mrow><mi>τ</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span> is smoother than that at <span><math><msup><mrow><mn>0</mn></mrow><mrow><mo>+</mo></mrow></msup></math></span>, where <span><math><mi>τ</mi></math></span> is a constant time delay, and this is an improvement for the work (Tan et al., 2022). Secondly, a high-order difference scheme based on <span><math><mrow><mi>L</mi><mn>1</mn></mrow></math></span> method is constructed. For the sake of repairing the convergence order in temporal direction and improving the computational efficiency, an efficient time two-grid algorithm based on nonuniform meshes is first developed. The convergence order of the two-grid scheme reaches <span><math><mrow><mi>O</mi><mrow><mo>(</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>F</mi></mrow><mrow><mo>−</mo><mo>min</mo><mrow><mo>{</mo><mi>r</mi><mi>α</mi><mo>,</mo><mn>2</mn><mo>−</mo><mi>α</mi><mo>}</mo></mrow></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi>C</mi></mrow><mrow><mo>−</mo><mo>min</mo><mrow><mo>{</mo><mn>2</mn><mi>r</mi><mi>α</mi><mo>,</mo><mn>4</mn><mo>−</mo><mn>2</mn><mi>α</mi><mo>}</mo></mrow></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>h</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>h</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>)</mo></mrow></mrow></math></span>, where <span><math><msub><mrow><mi>N</mi></mrow><mrow><mi>F</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>N</mi></mrow><mrow><mi>C</mi></mrow></msub></math></span> represent the number of the fine and coarse grids respectively, while <span><math><msub><mrow><mi>h</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>h</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> are the space-step sizes. Furthermore, stability and convergence analysis of the proposed scheme are carefully verified by energy method. Finally, numerical experiments are carried out to show the validity of theoretical statements.</p></div>\",\"PeriodicalId\":50658,\"journal\":{\"name\":\"Communications in Nonlinear Science and Numerical Simulation\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":3.4000,\"publicationDate\":\"2024-08-13\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Communications in Nonlinear Science and Numerical Simulation\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S1007570424004556\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications in Nonlinear Science and Numerical Simulation","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1007570424004556","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Variable-step L1 method combined with time two-grid algorithm for multi-singularity problems arising from two-dimensional nonlinear delay fractional equations
In this paper, we focus on the numerical simulation for two-dimensional nonlinear fractional sub-diffusion equations in the presence of time delay. Firstly, we investigate the existence, uniqueness and regularity of the solution for such problems. The theoretical result implies that the solution at is smoother than that at , where is a constant time delay, and this is an improvement for the work (Tan et al., 2022). Secondly, a high-order difference scheme based on method is constructed. For the sake of repairing the convergence order in temporal direction and improving the computational efficiency, an efficient time two-grid algorithm based on nonuniform meshes is first developed. The convergence order of the two-grid scheme reaches , where and represent the number of the fine and coarse grids respectively, while and are the space-step sizes. Furthermore, stability and convergence analysis of the proposed scheme are carefully verified by energy method. Finally, numerical experiments are carried out to show the validity of theoretical statements.
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The journal publishes original research findings on experimental observation, mathematical modeling, theoretical analysis and numerical simulation, for more accurate description, better prediction or novel application, of nonlinear phenomena in science and engineering. It offers a venue for researchers to make rapid exchange of ideas and techniques in nonlinear science and complexity.
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Nonlinear differential or delay equations, Lie group analysis and asymptotic methods, Discontinuous systems, Fractals, Fractional calculus and dynamics, Nonlinear effects in quantum mechanics, Nonlinear stochastic processes, Experimental nonlinear science, Time-series and signal analysis, Computational methods and simulations in nonlinear science and engineering, Control of dynamical systems, Synchronization, Lyapunov analysis, High-dimensional chaos and turbulence, Chaos in Hamiltonian systems, Integrable systems and solitons, Collective behavior in many-body systems, Biological physics and networks, Nonlinear mechanical systems, Complex systems and complexity.
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