Stability of step size control based on a posteriori error estimates
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866908058757627904 |
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| author | Ranocha, Hendrik Giesselmann, Jan |
| author_facet | Ranocha, Hendrik Giesselmann, Jan |
| contents | A posteriori error estimates based on residuals can be used for reliable error control of numerical methods. Here, we consider them in the context of ordinary differential equations and Runge-Kutta methods. In particular, we take the approach of Dedner & Giesselmann (2016) and investigate it when used to select the time step size. We focus on step size control stability when combined with explicit Runge-Kutta methods and demonstrate that a standard I controller is unstable while more advanced PI and PID controllers can be designed to be stable. We compare the stability properties of residual-based estimators and classical error estimators based on an embedded Runge-Kutta method both analytically and in numerical experiments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_12677 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Stability of step size control based on a posteriori error estimates Ranocha, Hendrik Giesselmann, Jan Numerical Analysis 65L06, 65M20 A posteriori error estimates based on residuals can be used for reliable error control of numerical methods. Here, we consider them in the context of ordinary differential equations and Runge-Kutta methods. In particular, we take the approach of Dedner & Giesselmann (2016) and investigate it when used to select the time step size. We focus on step size control stability when combined with explicit Runge-Kutta methods and demonstrate that a standard I controller is unstable while more advanced PI and PID controllers can be designed to be stable. We compare the stability properties of residual-based estimators and classical error estimators based on an embedded Runge-Kutta method both analytically and in numerical experiments. |
| title | Stability of step size control based on a posteriori error estimates |
| topic | Numerical Analysis 65L06, 65M20 |
| url | https://arxiv.org/abs/2307.12677 |