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Manchester Applied Mathematics and Numerical Analysis SeminarsWinter 1998 |
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Room 2.16 in the Maths Building, Oxford Road
Stability analysis of numerical methods for ordinary differential equations is motivated by the question ``for what choices of stepsize does the numerical method reproduce the characteristics of the test equation?'' We study a linear test equation with a multiplicative noise term, and consider mean-square and asymptotic stability of a stochastic version of the Theta Method. We extend some mean-square stability results in [Saito and Mitsui, SIAM. J. Numer. Anal., 33, 1996, 2254--2267]. In particular, we show that an extension of the deterministic A-stability property holds. We also plot mean-square stability regions for the case where the test equation has real parameters. For asymptotic stability, we show that the issue reduces to finding the expected value of a parametrized random variable. We combine analytical and numerical techniques to get insights into the stability properties. For a variant of the method that has been proposed in the literature we obtain precise analytic expressions for the asymptotic stability region. This allows us to prove a number of results. The technique introduced is widely applicable, and we use it to show that a fully implicit method suggested by Kloeden and Platen has an asymptotic stability extension of the deterministic A-stability property. We also use the approach to explain some numerical results reported in [Milstein, Platen and Schurz, SIAM J. Numer. Anal., 35, 1998, 1010--1019.]
For further info contact either Matthias Heil (mheil@ma.man.ac.uk), Mark Muldoon (M.Muldoon@umist.ac.uk)or the seminar secretary (Tel. 0161 275 5800).