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Multirate GARK schemes and infinitesimal extensions Sandu, Adrian


Differential equations arising in many practical applications are characterized by multiple time scales. Multirate time integration seeks to solve them efficiently by discretizing each scale with a different, appropriate time step, while ensuring the overall accuracy and stability of the numerical solution. While the multirate idea is elegant and has been around for decades, multirate methods are not yet widely used in applications. This is due, in part, to the difficulties raised by the construction of high-order multirate schemes. We discuss the design of practical high-order multirate methods using the theoretical framework of generalized additive Runge--Kutta methods MR-GARK, which provides the generic order conditions and the linear and nonlinear stability analyses. In a seminal paper Knoth and Wolke (1998) proposed a hybrid solution approach: discretize the slow component with an explicit Runge-Kutta method, and advance the fast component via a modified fast differential equation. The idea led to the development of multirate infinitesimal step (MIS) methods in (Wensch et al. 2009). Guenther and Sandu (2016) explained MIS schemes as a particular case of MR-GARK methods. The hybrid approach offers extreme flexibility in the choice of the numerical solution process for the fast component. This work discusses new families of multirate infinitesimal GARK schemes (MRI-GARK) that extends the MIS approach in multiple ways. Order conditions theory and stability analyses are developed, and practical explicit and implicit methods of up to order four are constructed. Numerical results confirm the theoretical findings. We expect the new MRI-GARK family to be most useful for systems of equations with widely disparate time scales, where the fast process is dispersive, and where the coupling between the fast and slow dynamics is relatively weak.

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