A Stiff Order Condition Theory for Runge-Kutta Methods Applied to Semilinear ODEs
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866914525302751232 |
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| author | Roberts, Steven B. Shirokoff, David Biswas, Abhijit Seibold, Benjamin |
| author_facet | Roberts, Steven B. Shirokoff, David Biswas, Abhijit Seibold, Benjamin |
| contents | Classical convergence theory of Runge-Kutta methods assumes that the time step is small relative to the Lipschitz constant of the ordinary differential equation (ODE). For stiff problems, that assumption is often violated, and a problematic degradation in accuracy, known as order reduction, can arise. Methods with high stage order, e.g., Gauss-Legendre and Radau, are known to avoid order reduction, but they must be fully implicit. For the broad class of semilinear ODEs, which consist of a stiff linear term and non-stiff nonlinear term, we show that weaker conditions suffice. Our new semilinear order conditions are formulated in terms of orthogonality relations and can be enumerated by rooted trees. Finally, we prove global error bounds that hold uniformly with respect to stiffness of the linear term. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_15099 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Stiff Order Condition Theory for Runge-Kutta Methods Applied to Semilinear ODEs Roberts, Steven B. Shirokoff, David Biswas, Abhijit Seibold, Benjamin Numerical Analysis Classical convergence theory of Runge-Kutta methods assumes that the time step is small relative to the Lipschitz constant of the ordinary differential equation (ODE). For stiff problems, that assumption is often violated, and a problematic degradation in accuracy, known as order reduction, can arise. Methods with high stage order, e.g., Gauss-Legendre and Radau, are known to avoid order reduction, but they must be fully implicit. For the broad class of semilinear ODEs, which consist of a stiff linear term and non-stiff nonlinear term, we show that weaker conditions suffice. Our new semilinear order conditions are formulated in terms of orthogonality relations and can be enumerated by rooted trees. Finally, we prove global error bounds that hold uniformly with respect to stiffness of the linear term. |
| title | A Stiff Order Condition Theory for Runge-Kutta Methods Applied to Semilinear ODEs |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2505.15099 |