Order conditions for partitioned Runge-Kutta methods

Abstract. Explicit time integration methods are characterised by a small numerical effort per time step. In the application to multiscale problems in atmospheric modelling, this benefit is often more than compensated by stability problems and step size restrictions resulting from stiff chemical reaction terms and from a locally varying Courant-FriedrichsLewy (CFL) condition for the advection terms. Splitting methods may be applied to efficiently combine implicit and explicit methods (IMEX splitting). Complementarily multirate time integration schemes allow for a local adaptation of the time step size to the grid size. In combination, these approaches lead to schemes which are efficient in terms of evaluations of the right-hand side. Special challenges arise when these methods are to be implemented. For an efficient implementation, it is crucial to locate and exploit redundancies. Furthermore, the more complex programme flow may lead to computational overhead which, in the worst case, more than compensates the theoretical gain in efficiency. We present a general splitting approach which allows both for IMEX splittings and for local time step adaptation. The main focus is on an efficient implementation of this approach for parallel computation on computer clusters.

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“…Order conditions for partitioned Runge-Kutta methods can be found for instance in Jackiewicz and Vermiglio (1998).…”

Section: Mathematical Preliminariesmentioning

confidence: 99%

Implementation of multirate time integration methods for air pollution modelling

et al. 2012

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“…Because this multirate strategy is based on the PRK method, the order of the coupled method can be obtained as the minimum among the base methods used and the order of their coupling [30,31]. This is a major advantage, in the context of multirate methods, compared with the standard continuous FEM where all types of elements are connected through nodes.…”

Section: Generalizationmentioning

confidence: 99%

“…They consist in both the classic order conditions for the base ERK methods and additional coupling conditions [31]. They consist in both the classic order conditions for the base ERK methods and additional coupling conditions [31].…”

Section: Recursive Flux Splitting Multiratementioning

confidence: 99%

“…However, we did not investigate in practice the multirate approach for continuous finite elements. Because this multirate strategy is based on the PRK method, the order of the coupled method can be obtained as the minimum among the base methods used and the order of their coupling [30,31]. Constantinescu [14] has shown that the MPRK-2 schemes, defined by the Butcher tableaus in Table III, are (1) second-order accurate if the base method is at least second-order accurate and (2) have at most a second-order accurate coupling regardless of the order of the base method.…”

Section: Generalizationmentioning

confidence: 99%

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Multirate time stepping for accelerating explicit discontinuous Galerkin computations with application to geophysical flows

Seny

Lambrechts

Comblen

et al. 2012

Numerical Methods in Fluids

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SUMMARY This paper presents multirate explicit time‐stepping schemes for solving partial differential equations with discontinuous Galerkin elements in the framework of Large‐scale marine flows. It addresses the variability of the local stable time steps by gathering the mesh elements in appropriate groups. The real challenge is to develop methods exhibiting mass conservation and consistency. Two multirate approaches, based on standard explicit Runge–Kutta methods, are analyzed. They are well suited and optimized for the discontinuous Galerkin framework. The significant speedups observed for the hydrodynamic application of the Great Barrier Reef confirm the theoretical expectations. Copyright © 2012 John Wiley & Sons, Ltd.

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“…As for explicit Runge-Kutta methods, formal order conditions can be derived [8]. These conditions consist of the order conditions for the individual Runge-Kutta methods and additional coupling conditions.…”

Section: Explicit Runge-kutta Multirate Scheme Based On a Right-hand mentioning

confidence: 99%

Multirate Runge–Kutta schemes for advection equations

Schlegel

Knoth

Arnold

et al. 2009

Journal of Computational and Applied Mathematics

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a b s t r a c tExplicit time integration methods can be employed to simulate a broad spectrum of physical phenomena. The wide range of scales encountered lead to the problem that the fastest cell of the simulation dictates the global time step. Multirate time integration methods can be employed to alter the time step locally so that slower components take longer and fewer time steps, resulting in a moderate to substantial reduction of the computational cost, depending on the scenario to simulate [S. Osher, R. Sanders, Numerical approximations to nonlinear conservation laws with locally varying time and space grids, Math. Comput. 41 (1983) 321-336; H. Tang, G. Warnecke, A class of high resolution schemes for hyperbolic conservation laws and convection-diffusion equations with varying time and pace grids, SIAM J. Sci. Comput. 26 (4) (2005) 1415-1431; E. Constantinescu, A. Sandu, Multirate timestepping methods for hyperbolic conservation laws, SIAM J. Sci. Comput. 33 (3) (2007) 239-278].In air pollution modeling the advection part is usually integrated explicitly in time, where the time step is constrained by a locally varying Courant-Friedrichs-Lewy (CFL) number. Multirate schemes are a useful tool to decouple different physical regions so that this constraint becomes a local instead of a global restriction. Therefore it is of major interest to apply multirate schemes to the advection equation. We introduce a generic recursive multirate Runge-Kutta scheme that can be easily adapted to an arbitrary number of refinement levels. It preserves the linear invariants of the system and is of third order accuracy when applied to certain explicit Runge-Kutta methods as base method.

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Order conditions for partitioned Runge-Kutta methods

Cited by 14 publications

References 9 publications

Implementation of multirate time integration methods for air pollution modelling

Implementation of multirate time integration methods for air pollution modelling

Multirate time stepping for accelerating explicit discontinuous Galerkin computations with application to geophysical flows

Multirate Runge–Kutta schemes for advection equations

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