Contrast-Independent, Partially-Explicit Time Discretizations for Nonlinear Multiscale Problems
This work continues a line of work on developing partially explicit methods for multiscale problems. In our previous works, we considered linear multiscale problems where the spatial heterogeneities are at the subgrid level and are not resolved. In these works, we have introduced contrast-independen...
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MDPI AG
2021-11-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/23/3000 |
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| author | Eric T. Chung Yalchin Efendiev Wing Tat Leung Wenyuan Li |
| author_facet | Eric T. Chung Yalchin Efendiev Wing Tat Leung Wenyuan Li |
| author_sort | Eric T. Chung |
| collection | DOAJ |
| description | This work continues a line of work on developing partially explicit methods for multiscale problems. In our previous works, we considered linear multiscale problems where the spatial heterogeneities are at the subgrid level and are not resolved. In these works, we have introduced contrast-independent, partially explicit time discretizations for linear equations. The contrast-independent, partially explicit time discretization divides the spatial space into two components: contrast dependent (fast) and contrast independent (slow) spaces defined via multiscale space decomposition. Following this decomposition, temporal splitting was proposed, which treats fast components implicitly and slow components explicitly. The space decomposition and temporal splitting are chosen such that they guarantees stability, and we formulated a condition for the time stepping. This condition was formulated as a condition on slow spaces. In this paper, we extend this approach to nonlinear problems. We propose a splitting approach and derive a condition that guarantees stability. This condition requires some type of contrast-independent spaces for slow components of the solution. We present numerical results and show that the proposed methods provide results similar to implicit methods with a time step that is independent of the contrast. |
| first_indexed | 2024-03-10T04:48:46Z |
| format | Article |
| id | doaj.art-cf32e06fa7d44df789616c5c77cad99e |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T04:48:46Z |
| publishDate | 2021-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-cf32e06fa7d44df789616c5c77cad99e2023-11-23T02:44:30ZengMDPI AGMathematics2227-73902021-11-01923300010.3390/math9233000Contrast-Independent, Partially-Explicit Time Discretizations for Nonlinear Multiscale ProblemsEric T. Chung0Yalchin Efendiev1Wing Tat Leung2Wenyuan Li3Department of Mathematics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong, ChinaDepartment of Mathematics, Institute for Scientific Computation (ISC), Texas A&M University, College Station, TX 77845, USADepartment of Mathematics, University of California, Irvine, CA 92697, USADepartment of Mathematics, Institute for Scientific Computation (ISC), Texas A&M University, College Station, TX 77845, USAThis work continues a line of work on developing partially explicit methods for multiscale problems. In our previous works, we considered linear multiscale problems where the spatial heterogeneities are at the subgrid level and are not resolved. In these works, we have introduced contrast-independent, partially explicit time discretizations for linear equations. The contrast-independent, partially explicit time discretization divides the spatial space into two components: contrast dependent (fast) and contrast independent (slow) spaces defined via multiscale space decomposition. Following this decomposition, temporal splitting was proposed, which treats fast components implicitly and slow components explicitly. The space decomposition and temporal splitting are chosen such that they guarantees stability, and we formulated a condition for the time stepping. This condition was formulated as a condition on slow spaces. In this paper, we extend this approach to nonlinear problems. We propose a splitting approach and derive a condition that guarantees stability. This condition requires some type of contrast-independent spaces for slow components of the solution. We present numerical results and show that the proposed methods provide results similar to implicit methods with a time step that is independent of the contrast.https://www.mdpi.com/2227-7390/9/23/3000multiscale methodGMsFEMsplittingnonlinear reactionCEM-GMsFEMexplicit–implicit |
| spellingShingle | Eric T. Chung Yalchin Efendiev Wing Tat Leung Wenyuan Li Contrast-Independent, Partially-Explicit Time Discretizations for Nonlinear Multiscale Problems Mathematics multiscale method GMsFEM splitting nonlinear reaction CEM-GMsFEM explicit–implicit |
| title | Contrast-Independent, Partially-Explicit Time Discretizations for Nonlinear Multiscale Problems |
| title_full | Contrast-Independent, Partially-Explicit Time Discretizations for Nonlinear Multiscale Problems |
| title_fullStr | Contrast-Independent, Partially-Explicit Time Discretizations for Nonlinear Multiscale Problems |
| title_full_unstemmed | Contrast-Independent, Partially-Explicit Time Discretizations for Nonlinear Multiscale Problems |
| title_short | Contrast-Independent, Partially-Explicit Time Discretizations for Nonlinear Multiscale Problems |
| title_sort | contrast independent partially explicit time discretizations for nonlinear multiscale problems |
| topic | multiscale method GMsFEM splitting nonlinear reaction CEM-GMsFEM explicit–implicit |
| url | https://www.mdpi.com/2227-7390/9/23/3000 |
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