A Multi-Parameter Singular Perturbation Analysis of the Robertson Model
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929699594174464 |
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| author | Baumgartner, Lukas Szmolyan, Peter |
| author_facet | Baumgartner, Lukas Szmolyan, Peter |
| contents | The Robertson model describing a chemical reaction involving three reactants is one of the classical examples of stiffness in ODEs. The stiffness is caused by the occurrence of three reaction rates $k_1,\,k_2$, and $k_3$, with largely differing orders of magnitude, acting as parameters. The model has been widely used as a numerical test problem. Surprisingly, no asymptotic analysis of this multiscale problem seems to exist. In this paper we provide a full asymptotic analysis of the Robertson model under the assumption $k_1, k_3 \ll k_2$. We rewrite the equations as a two-parameter singular perturbation problem in the rescaled small parameters $(\varepsilon_1,\varepsilon_2):=(k_1/k_2,k_3/k_2)$, which we then analyze using geometric singular perturbation theory (GSPT). To deal with the multi-parameter singular structure, we perform blow-ups in parameter- and variable space. We identify four distinct regimes in a neighbourhood of the singular limit $(\varepsilon_1,\varepsilon_2)= (0,0)$. Within these four regimes we use GSPT and additional blow-ups to analyze the dynamics and the structure of solutions. Our asymptotic results are in excellent qualitative and quantitative agreement with the numerics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_04008 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Multi-Parameter Singular Perturbation Analysis of the Robertson Model Baumgartner, Lukas Szmolyan, Peter Dynamical Systems Classical Analysis and ODEs 34E10, 34E13, 34E15, 92E20 The Robertson model describing a chemical reaction involving three reactants is one of the classical examples of stiffness in ODEs. The stiffness is caused by the occurrence of three reaction rates $k_1,\,k_2$, and $k_3$, with largely differing orders of magnitude, acting as parameters. The model has been widely used as a numerical test problem. Surprisingly, no asymptotic analysis of this multiscale problem seems to exist. In this paper we provide a full asymptotic analysis of the Robertson model under the assumption $k_1, k_3 \ll k_2$. We rewrite the equations as a two-parameter singular perturbation problem in the rescaled small parameters $(\varepsilon_1,\varepsilon_2):=(k_1/k_2,k_3/k_2)$, which we then analyze using geometric singular perturbation theory (GSPT). To deal with the multi-parameter singular structure, we perform blow-ups in parameter- and variable space. We identify four distinct regimes in a neighbourhood of the singular limit $(\varepsilon_1,\varepsilon_2)= (0,0)$. Within these four regimes we use GSPT and additional blow-ups to analyze the dynamics and the structure of solutions. Our asymptotic results are in excellent qualitative and quantitative agreement with the numerics. |
| title | A Multi-Parameter Singular Perturbation Analysis of the Robertson Model |
| topic | Dynamical Systems Classical Analysis and ODEs 34E10, 34E13, 34E15, 92E20 |
| url | https://arxiv.org/abs/2407.04008 |