Algebraic Conditions for Stability in Runge-Kutta Methods and Their Certification via Semidefinite Programming
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| Format: | Preprint |
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2024
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| _version_ | 1866914807456727040 |
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| author | Juhl, Austin Shirokoff, David |
| author_facet | Juhl, Austin Shirokoff, David |
| contents | In this work, we present approaches to rigorously certify $A$- and $A(α)$-stability in Runge-Kutta methods through the solution of convex feasibility problems defined by linear matrix inequalities. We adopt two approaches. The first is based on sum-of-squares programming applied to the Runge-Kutta $E$-polynomial and is applicable to both $A$- and $A(α)$-stability. In the second, we sharpen the algebraic conditions for $A$-stability of Cooper, Scherer, T{ü}rke, and Wendler to incorporate the Runge-Kutta order conditions. We demonstrate how the theoretical improvement enables the practical use of these conditions for certification of $A$-stability within a computational framework. We then use both approaches to obtain rigorous certificates of stability for several diagonally implicit schemes devised in the literature. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_13921 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Algebraic Conditions for Stability in Runge-Kutta Methods and Their Certification via Semidefinite Programming Juhl, Austin Shirokoff, David Numerical Analysis Optimization and Control 65L06, 65L07, 65L20 In this work, we present approaches to rigorously certify $A$- and $A(α)$-stability in Runge-Kutta methods through the solution of convex feasibility problems defined by linear matrix inequalities. We adopt two approaches. The first is based on sum-of-squares programming applied to the Runge-Kutta $E$-polynomial and is applicable to both $A$- and $A(α)$-stability. In the second, we sharpen the algebraic conditions for $A$-stability of Cooper, Scherer, T{ü}rke, and Wendler to incorporate the Runge-Kutta order conditions. We demonstrate how the theoretical improvement enables the practical use of these conditions for certification of $A$-stability within a computational framework. We then use both approaches to obtain rigorous certificates of stability for several diagonally implicit schemes devised in the literature. |
| title | Algebraic Conditions for Stability in Runge-Kutta Methods and Their Certification via Semidefinite Programming |
| topic | Numerical Analysis Optimization and Control 65L06, 65L07, 65L20 |
| url | https://arxiv.org/abs/2405.13921 |