Application of Miller's theorem to the stability analysis of numerical schemes; some useful tools for rapid inspection of discretisations in ocean modelling |
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| Institution: | 1. Department of Mathematics Sciences, New Jersey Institute of Technology, Newark, NJ 07102, USA;2. Computational Science Research Center, Beijing 100193, China;3. Department of Mathematics and Statistics, Old Dominion University, Norfolk, VA 23529, USA |
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| Abstract: | An old theorem about the roots of a polynomial is resurfaced and examined, focusing on the practical question of determining stability criteria for numerical schemes. Is is demonstrated that the theorem can be useful both for analytical studies of the stability limits, as well as numerical searches for stability regions. It is particularly important that when deciding whether the scheme is stable or not, it is not necessary to search for the roots of the polynomial. Indeed, such a decision may be reached through a finite and small number of arithmetic operations and verifications of inequalities. In addition to the analysis of the theorem, some practical conditions for polynomial of different orders are presented, as well as some useful tips on how simply necessary or sufficient stability conditions can be obtained. |
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