Construction and Implementation of Efficient Numerical Methods for Ordinary Differential Equations

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Many problems in science and engineering are modelled by large systems of differential equations. It is the purpose of the research described in this proposal to design and implement accurate, efficient, reliable, and robust methods for the numerical simulations of such systems. These methods should be capable of exploiting special structure often present in such systems to increase the computational efficiency. Examples of such systems with special structure include many problems in computational fluid dynamics, molecular biology, quantum mechanics, and heat transfer, and their numerical solution requires new more powerful numerical techniques than the classical methods. The novel techniques which will be investigated in this proposal are based on the recent theory of general linear methods and on exponential integrators which treat differently specific parts of the differential systems. Some of the proposed schemes are appropriate for implementation in a parallel computing environment.