Asymptotic Behaviour of Time Stepping Methods for Phase Field Models
Abstract Adaptive time stepping methods for metastable dynamics of the Allen–Cahn and Cahn–Hilliard equations are investigated in the spatially continuous, semi-discrete setting. We analyse the performance of a number of first and second order methods, formally predicting step sizes required to sati...
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| Autor*in: |
Cheng, Xinyu [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2021 |
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| Schlagwörter: |
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| Anmerkung: |
© The Author(s) 2021 |
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| Übergeordnetes Werk: |
Enthalten in: Journal of scientific computing - Springer US, 1986, 86(2021), 3 vom: 16. Jan. |
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| Übergeordnetes Werk: |
volume:86 ; year:2021 ; number:3 ; day:16 ; month:01 |
| Links: |
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| DOI / URN: |
10.1007/s10915-020-01391-x |
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OLC2122785322 |
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| 520 | |a Abstract Adaptive time stepping methods for metastable dynamics of the Allen–Cahn and Cahn–Hilliard equations are investigated in the spatially continuous, semi-discrete setting. We analyse the performance of a number of first and second order methods, formally predicting step sizes required to satisfy specified local truncation error $$\sigma $$ in the limit of small length scale parameter $$\epsilon \rightarrow 0$$ during meta-stable dynamics. The formal predictions are made under stability assumptions that include the preservation of the asymptotic structure of the diffuse interface, a concept we call profile fidelity. In this setting, definite statements about the relative behaviour of time stepping methods can be made. Some methods, including all so-called energy stable methods but also some fully implicit methods, require asymptotically more time steps than others. The formal analysis is confirmed in computational studies. We observe that some provably energy stable methods popular in the literature perform worse than some more standard schemes. We show further that when Backward Euler is applied to meta-stable Allen–Cahn dynamics, the energy decay and profile fidelity properties for these discretizations are preserved for much larger time steps than previous analysis would suggest. The results are established asymptotically for general interfaces, with a rigorous proof for radial interfaces. It is shown analytically and computationally that for most reaction terms, Eyre type time stepping performs asymptotically worse due to loss of profile fidelity. | ||
| 650 | 4 | |a Allen–Cahn equation | |
| 650 | 4 | |a Cahn–Hilliard equation | |
| 650 | 4 | |a Phase field model | |
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| 650 | 4 | |a Energy stability | |
| 700 | 1 | |a Li, Dong |4 aut | |
| 700 | 1 | |a Promislow, Keith |4 aut | |
| 700 | 1 | |a Wetton, Brian |0 (orcid)0000-0002-6808-6301 |4 aut | |
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10.1007/s10915-020-01391-x doi (DE-627)OLC2122785322 (DE-He213)s10915-020-01391-x-p DE-627 ger DE-627 rakwb eng 004 VZ 11 ssgn Cheng, Xinyu verfasserin aut Asymptotic Behaviour of Time Stepping Methods for Phase Field Models 2021 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s) 2021 Abstract Adaptive time stepping methods for metastable dynamics of the Allen–Cahn and Cahn–Hilliard equations are investigated in the spatially continuous, semi-discrete setting. We analyse the performance of a number of first and second order methods, formally predicting step sizes required to satisfy specified local truncation error $$\sigma $$ in the limit of small length scale parameter $$\epsilon \rightarrow 0$$ during meta-stable dynamics. The formal predictions are made under stability assumptions that include the preservation of the asymptotic structure of the diffuse interface, a concept we call profile fidelity. In this setting, definite statements about the relative behaviour of time stepping methods can be made. Some methods, including all so-called energy stable methods but also some fully implicit methods, require asymptotically more time steps than others. The formal analysis is confirmed in computational studies. We observe that some provably energy stable methods popular in the literature perform worse than some more standard schemes. We show further that when Backward Euler is applied to meta-stable Allen–Cahn dynamics, the energy decay and profile fidelity properties for these discretizations are preserved for much larger time steps than previous analysis would suggest. The results are established asymptotically for general interfaces, with a rigorous proof for radial interfaces. It is shown analytically and computationally that for most reaction terms, Eyre type time stepping performs asymptotically worse due to loss of profile fidelity. Allen–Cahn equation Cahn–Hilliard equation Phase field model Time stepping Energy stability Li, Dong aut Promislow, Keith aut Wetton, Brian (orcid)0000-0002-6808-6301 aut Enthalten in Journal of scientific computing Springer US, 1986 86(2021), 3 vom: 16. Jan. (DE-627)129217549 (DE-600)56055-8 (DE-576)065121945 0885-7474 nnns volume:86 year:2021 number:3 day:16 month:01 https://doi.org/10.1007/s10915-020-01391-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT AR 86 2021 3 16 01 |
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10.1007/s10915-020-01391-x doi (DE-627)OLC2122785322 (DE-He213)s10915-020-01391-x-p DE-627 ger DE-627 rakwb eng 004 VZ 11 ssgn Cheng, Xinyu verfasserin aut Asymptotic Behaviour of Time Stepping Methods for Phase Field Models 2021 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s) 2021 Abstract Adaptive time stepping methods for metastable dynamics of the Allen–Cahn and Cahn–Hilliard equations are investigated in the spatially continuous, semi-discrete setting. We analyse the performance of a number of first and second order methods, formally predicting step sizes required to satisfy specified local truncation error $$\sigma $$ in the limit of small length scale parameter $$\epsilon \rightarrow 0$$ during meta-stable dynamics. The formal predictions are made under stability assumptions that include the preservation of the asymptotic structure of the diffuse interface, a concept we call profile fidelity. In this setting, definite statements about the relative behaviour of time stepping methods can be made. Some methods, including all so-called energy stable methods but also some fully implicit methods, require asymptotically more time steps than others. The formal analysis is confirmed in computational studies. We observe that some provably energy stable methods popular in the literature perform worse than some more standard schemes. We show further that when Backward Euler is applied to meta-stable Allen–Cahn dynamics, the energy decay and profile fidelity properties for these discretizations are preserved for much larger time steps than previous analysis would suggest. The results are established asymptotically for general interfaces, with a rigorous proof for radial interfaces. It is shown analytically and computationally that for most reaction terms, Eyre type time stepping performs asymptotically worse due to loss of profile fidelity. Allen–Cahn equation Cahn–Hilliard equation Phase field model Time stepping Energy stability Li, Dong aut Promislow, Keith aut Wetton, Brian (orcid)0000-0002-6808-6301 aut Enthalten in Journal of scientific computing Springer US, 1986 86(2021), 3 vom: 16. Jan. (DE-627)129217549 (DE-600)56055-8 (DE-576)065121945 0885-7474 nnns volume:86 year:2021 number:3 day:16 month:01 https://doi.org/10.1007/s10915-020-01391-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT AR 86 2021 3 16 01 |
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10.1007/s10915-020-01391-x doi (DE-627)OLC2122785322 (DE-He213)s10915-020-01391-x-p DE-627 ger DE-627 rakwb eng 004 VZ 11 ssgn Cheng, Xinyu verfasserin aut Asymptotic Behaviour of Time Stepping Methods for Phase Field Models 2021 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s) 2021 Abstract Adaptive time stepping methods for metastable dynamics of the Allen–Cahn and Cahn–Hilliard equations are investigated in the spatially continuous, semi-discrete setting. We analyse the performance of a number of first and second order methods, formally predicting step sizes required to satisfy specified local truncation error $$\sigma $$ in the limit of small length scale parameter $$\epsilon \rightarrow 0$$ during meta-stable dynamics. The formal predictions are made under stability assumptions that include the preservation of the asymptotic structure of the diffuse interface, a concept we call profile fidelity. In this setting, definite statements about the relative behaviour of time stepping methods can be made. Some methods, including all so-called energy stable methods but also some fully implicit methods, require asymptotically more time steps than others. The formal analysis is confirmed in computational studies. We observe that some provably energy stable methods popular in the literature perform worse than some more standard schemes. We show further that when Backward Euler is applied to meta-stable Allen–Cahn dynamics, the energy decay and profile fidelity properties for these discretizations are preserved for much larger time steps than previous analysis would suggest. The results are established asymptotically for general interfaces, with a rigorous proof for radial interfaces. It is shown analytically and computationally that for most reaction terms, Eyre type time stepping performs asymptotically worse due to loss of profile fidelity. Allen–Cahn equation Cahn–Hilliard equation Phase field model Time stepping Energy stability Li, Dong aut Promislow, Keith aut Wetton, Brian (orcid)0000-0002-6808-6301 aut Enthalten in Journal of scientific computing Springer US, 1986 86(2021), 3 vom: 16. Jan. (DE-627)129217549 (DE-600)56055-8 (DE-576)065121945 0885-7474 nnns volume:86 year:2021 number:3 day:16 month:01 https://doi.org/10.1007/s10915-020-01391-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT AR 86 2021 3 16 01 |
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10.1007/s10915-020-01391-x doi (DE-627)OLC2122785322 (DE-He213)s10915-020-01391-x-p DE-627 ger DE-627 rakwb eng 004 VZ 11 ssgn Cheng, Xinyu verfasserin aut Asymptotic Behaviour of Time Stepping Methods for Phase Field Models 2021 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s) 2021 Abstract Adaptive time stepping methods for metastable dynamics of the Allen–Cahn and Cahn–Hilliard equations are investigated in the spatially continuous, semi-discrete setting. We analyse the performance of a number of first and second order methods, formally predicting step sizes required to satisfy specified local truncation error $$\sigma $$ in the limit of small length scale parameter $$\epsilon \rightarrow 0$$ during meta-stable dynamics. The formal predictions are made under stability assumptions that include the preservation of the asymptotic structure of the diffuse interface, a concept we call profile fidelity. In this setting, definite statements about the relative behaviour of time stepping methods can be made. Some methods, including all so-called energy stable methods but also some fully implicit methods, require asymptotically more time steps than others. The formal analysis is confirmed in computational studies. We observe that some provably energy stable methods popular in the literature perform worse than some more standard schemes. We show further that when Backward Euler is applied to meta-stable Allen–Cahn dynamics, the energy decay and profile fidelity properties for these discretizations are preserved for much larger time steps than previous analysis would suggest. The results are established asymptotically for general interfaces, with a rigorous proof for radial interfaces. It is shown analytically and computationally that for most reaction terms, Eyre type time stepping performs asymptotically worse due to loss of profile fidelity. Allen–Cahn equation Cahn–Hilliard equation Phase field model Time stepping Energy stability Li, Dong aut Promislow, Keith aut Wetton, Brian (orcid)0000-0002-6808-6301 aut Enthalten in Journal of scientific computing Springer US, 1986 86(2021), 3 vom: 16. Jan. (DE-627)129217549 (DE-600)56055-8 (DE-576)065121945 0885-7474 nnns volume:86 year:2021 number:3 day:16 month:01 https://doi.org/10.1007/s10915-020-01391-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT AR 86 2021 3 16 01 |
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10.1007/s10915-020-01391-x doi (DE-627)OLC2122785322 (DE-He213)s10915-020-01391-x-p DE-627 ger DE-627 rakwb eng 004 VZ 11 ssgn Cheng, Xinyu verfasserin aut Asymptotic Behaviour of Time Stepping Methods for Phase Field Models 2021 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s) 2021 Abstract Adaptive time stepping methods for metastable dynamics of the Allen–Cahn and Cahn–Hilliard equations are investigated in the spatially continuous, semi-discrete setting. We analyse the performance of a number of first and second order methods, formally predicting step sizes required to satisfy specified local truncation error $$\sigma $$ in the limit of small length scale parameter $$\epsilon \rightarrow 0$$ during meta-stable dynamics. The formal predictions are made under stability assumptions that include the preservation of the asymptotic structure of the diffuse interface, a concept we call profile fidelity. In this setting, definite statements about the relative behaviour of time stepping methods can be made. Some methods, including all so-called energy stable methods but also some fully implicit methods, require asymptotically more time steps than others. The formal analysis is confirmed in computational studies. We observe that some provably energy stable methods popular in the literature perform worse than some more standard schemes. We show further that when Backward Euler is applied to meta-stable Allen–Cahn dynamics, the energy decay and profile fidelity properties for these discretizations are preserved for much larger time steps than previous analysis would suggest. The results are established asymptotically for general interfaces, with a rigorous proof for radial interfaces. It is shown analytically and computationally that for most reaction terms, Eyre type time stepping performs asymptotically worse due to loss of profile fidelity. Allen–Cahn equation Cahn–Hilliard equation Phase field model Time stepping Energy stability Li, Dong aut Promislow, Keith aut Wetton, Brian (orcid)0000-0002-6808-6301 aut Enthalten in Journal of scientific computing Springer US, 1986 86(2021), 3 vom: 16. Jan. (DE-627)129217549 (DE-600)56055-8 (DE-576)065121945 0885-7474 nnns volume:86 year:2021 number:3 day:16 month:01 https://doi.org/10.1007/s10915-020-01391-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT AR 86 2021 3 16 01 |
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Abstract Adaptive time stepping methods for metastable dynamics of the Allen–Cahn and Cahn–Hilliard equations are investigated in the spatially continuous, semi-discrete setting. We analyse the performance of a number of first and second order methods, formally predicting step sizes required to satisfy specified local truncation error $$\sigma $$ in the limit of small length scale parameter $$\epsilon \rightarrow 0$$ during meta-stable dynamics. The formal predictions are made under stability assumptions that include the preservation of the asymptotic structure of the diffuse interface, a concept we call profile fidelity. In this setting, definite statements about the relative behaviour of time stepping methods can be made. Some methods, including all so-called energy stable methods but also some fully implicit methods, require asymptotically more time steps than others. The formal analysis is confirmed in computational studies. We observe that some provably energy stable methods popular in the literature perform worse than some more standard schemes. We show further that when Backward Euler is applied to meta-stable Allen–Cahn dynamics, the energy decay and profile fidelity properties for these discretizations are preserved for much larger time steps than previous analysis would suggest. The results are established asymptotically for general interfaces, with a rigorous proof for radial interfaces. It is shown analytically and computationally that for most reaction terms, Eyre type time stepping performs asymptotically worse due to loss of profile fidelity. © The Author(s) 2021 |
| abstractGer |
Abstract Adaptive time stepping methods for metastable dynamics of the Allen–Cahn and Cahn–Hilliard equations are investigated in the spatially continuous, semi-discrete setting. We analyse the performance of a number of first and second order methods, formally predicting step sizes required to satisfy specified local truncation error $$\sigma $$ in the limit of small length scale parameter $$\epsilon \rightarrow 0$$ during meta-stable dynamics. The formal predictions are made under stability assumptions that include the preservation of the asymptotic structure of the diffuse interface, a concept we call profile fidelity. In this setting, definite statements about the relative behaviour of time stepping methods can be made. Some methods, including all so-called energy stable methods but also some fully implicit methods, require asymptotically more time steps than others. The formal analysis is confirmed in computational studies. We observe that some provably energy stable methods popular in the literature perform worse than some more standard schemes. We show further that when Backward Euler is applied to meta-stable Allen–Cahn dynamics, the energy decay and profile fidelity properties for these discretizations are preserved for much larger time steps than previous analysis would suggest. The results are established asymptotically for general interfaces, with a rigorous proof for radial interfaces. It is shown analytically and computationally that for most reaction terms, Eyre type time stepping performs asymptotically worse due to loss of profile fidelity. © The Author(s) 2021 |
| abstract_unstemmed |
Abstract Adaptive time stepping methods for metastable dynamics of the Allen–Cahn and Cahn–Hilliard equations are investigated in the spatially continuous, semi-discrete setting. We analyse the performance of a number of first and second order methods, formally predicting step sizes required to satisfy specified local truncation error $$\sigma $$ in the limit of small length scale parameter $$\epsilon \rightarrow 0$$ during meta-stable dynamics. The formal predictions are made under stability assumptions that include the preservation of the asymptotic structure of the diffuse interface, a concept we call profile fidelity. In this setting, definite statements about the relative behaviour of time stepping methods can be made. Some methods, including all so-called energy stable methods but also some fully implicit methods, require asymptotically more time steps than others. The formal analysis is confirmed in computational studies. We observe that some provably energy stable methods popular in the literature perform worse than some more standard schemes. We show further that when Backward Euler is applied to meta-stable Allen–Cahn dynamics, the energy decay and profile fidelity properties for these discretizations are preserved for much larger time steps than previous analysis would suggest. The results are established asymptotically for general interfaces, with a rigorous proof for radial interfaces. It is shown analytically and computationally that for most reaction terms, Eyre type time stepping performs asymptotically worse due to loss of profile fidelity. © The Author(s) 2021 |
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Asymptotic Behaviour of Time Stepping Methods for Phase Field Models |
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Li, Dong Promislow, Keith Wetton, Brian |
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2024-07-03T14:44:06.853Z |
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