Symplectic multirate generalized additive Runge-Kutta methods for Hamiltonian systems
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Generalized additive Runge-Kutta (GARK) schemes have shown to be a suitable tool for solving ordinary differential equations with additively partitioned right-hand sides. This work combines the ideas of symplectic GARK schemes and multirate GARK schemes to solve additively partitioned Hamiltonian systems with multirate behavior more efficiently. In a general setting of non-separable Hamiltonian systems, we derive order conditions, as well as conditions for symplecticity and time-reversibility. Moreover, investigations of the special case of separable Hamiltonian systems are carried out. We show that particular partitions may introduce stability issues and point out partitions that enable an implicit-explicit integration that comes with improved stability properties. Higher-order methods based on advanced composition techniques are discussed. Numerical results for the Fermi-Pasta-Ulam problem underline the performance of the schemes.
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