Implicit multirate GARK methods

Abstract

This work considers multirate generalized-structure additively partitioned Runge-Kutta (MrGARK) methods for solving stiff systems of ordinary differential equations (ODEs) with multiple time scales. These methods treat different partitions of the system with different timesteps for a more targeted and efficient solution compared to monolithic single rate approaches. With implicit methods used across all partitions, methods must find a balance between stability and the cost of solving nonlinear equations for the stages. In order to characterize this important trade-off, we explore multirate coupling strategies, problems for assessing linear stability, and techniques to efficiently implement Newton iterations for stage equations. Unlike much of the existing multirate stability analysis which is limited in scope to particular methods, we present general statements on stability and describe fundamental limitations for certain types of multirate schemes. New implicit multirate methods up to fourth order are derived, and their accuracy and efficiency properties are verified with numerical tests.

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Cited by in corpus (8)

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• Implicit-explicit multirate infinitesimal GARK methods
• Implicit-Explicit Multirate Infinitesimal Stage-Restart Methods
• A fast time-stepping strategy for dynamical systems equipped with a surrogate model
• High-Order Multirate Explicit Time-Stepping Schemes for the Baroclinic-Barotropic Split Dynamics in Primitive Equations
• Performance of explicit and IMEX MRI multirate methods on complex reactive flow problems within modern parallel adaptive structured grid frameworks
• Coupled Multirate Infinitesimal GARK Schemes for Stiff Systems with Multiple Time Scales
• A unified formulation of splitting-based implicit time integration schemes