Discrete Multirate GARK schems

Multirate time integration schemes apply different step sizes to different components of a system based on the local dynamics of the components. Local selection of step sizes allows increased computational efficiency while maintaining the desired solution accuracy. The multirating idea is elegant and has been around for decades, however, difficulties faced in construction of high order multirate schemes has hampered their application. Seeking to overcome these challenges, our work focuses on the design of high-order multirate methods using the theoretical framework of generalized additive Runge-Kutta (GARK) methods, which provides the generic order conditions and the stability analyses. Of special interest is deriving methods that avoid unnecessary coupling between the components of the system, and allow straightforward transition to different step sizes between the steps. We present Multirate GARK schemes of up to order four that are explicit-explicit, implicit-explicit, and explicit-implicit in different components. We present numerical experiments illustrating the performance of these new schemes.

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