A Generalized-Structure Approach to Additive Runge--Kutta Methods

Sandu, Adrian; Günther, Michael

doi:10.1137/130943224

Cited by 63 publications

(93 citation statements)

References 19 publications

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“…with the corresponding Butcher table [25,26] extended the ARK schemes to the so-called generalized-structure additively partitioned Runge-Kutta (GARK) schemes. The most general form of a GARK scheme is given by the following definition.…”

Section: Partitioning Systems Of Ordinary Differential Equationsmentioning

confidence: 99%

“…The order conditions of GARK schemes have been explicitly derived by Sandu and Günther up to schemes of 4th order, see [25]. The derivation is based on the application of the theory of N-trees by Araújo et al [30] on the representation of the GARK scheme in Equations (5) and (6).…”

Section: Order Conditions For Mgark Schemes With Variable Micro Stepsmentioning

confidence: 99%

“…Very recently, Sandu and Günther [25] constructed a generalized-structure approach to additively partitioned Runge-Kutta (GARK) methods. This approach comes with a high level of flexibility and allows for different Runge-Kutta stage values as arguments of different components of the problem's additively partitioned right-hand side.…”

Section: Introductionmentioning

confidence: 99%

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On Multirate GARK Schemes with Adaptive Micro Step Sizes for Fluid–Structure Interaction: Order Conditions and Preservation of the Geometric Conservation Law

Bremicker-Trübelhorn

Ortleb

2017

Aerospace

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Abstract:The application of partitioned schemes to fluid-structure interaction (FSI) allows the use of already developed solvers specifically designed for the efficient solution of the corresponding subproblems. In this work, we propose and describe a loosely coupled partitioned scheme based on the recently introduced generalized-structure additively partitioned Runge-Kutta (GARK) framework. The resulting scheme combines implicit-explicit (IMEX) and multirate approaches while coupling of the subproblems is realized both on the level of the discrete time steps and at the level of interior Runge-Kutta stages. Specifically, we allow for varying micro step sizes for the fluid subproblem and therefore extend the multirate GARK framework based on constant micro steps. Furthermore, we derive the order conditions for this extension allowing for coupled time integration schemes of up to third order and discuss specific choices of the Runge-Kutta coefficients complying with the geometric conservation law. Finally, numerical experiments are carried out for uniform flow on a moving grid as well as the classical FSI test case of a moving piston.

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Section: Partitioning Systems Of Ordinary Differential Equationsmentioning

confidence: 99%

Section: Order Conditions For Mgark Schemes With Variable Micro Stepsmentioning

confidence: 99%

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On Multirate GARK Schemes with Adaptive Micro Step Sizes for Fluid–Structure Interaction: Order Conditions and Preservation of the Geometric Conservation Law

Bremicker-Trübelhorn

Ortleb

2017

Aerospace

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“…This section investigates the conditions under which the multirate scheme (6a, 6b, 6c) is nonlinearly stable, i.e. the inequality y n+1 −ỹ n+1 ≤ y n −ỹ n holds for any two numerical approximations y n+1 and y n+1 obtained by applying the scheme to the ODE (1) with (11) and with initial values y n and y n .…”

Section: Nonlinear Stabilitymentioning

confidence: 99%

“…Generalized additive Runge-Kutta (GARK) methods were introduced in [11] to solve initial value problems for additively partitioned systems of ordinary differential equations …”

Section: Introductionmentioning

confidence: 99%

Multirate generalized additive Runge Kutta methods

Günther

Sandu

2015

Numer. Math.

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This work constructs a new class of multirate schemes based on the recently developed generalized additive Runge-Kutta (GARK) methods (Sandu and Günther, SIAM J Numer Anal, 53(1): 2015). Multirate schemes use different step sizes for different components and for different partitions of the right-hand side based on the local activity levels. We show that the new multirate GARK family includes many wellknown multirate schemes as special cases. The order conditions theory follows directly from the GARK accuracy theory. Nonlinear stability and monotonicity investigations show that these properties are inherited from the base schemes provided that additional coupling conditions hold.Mathematics Subject Classification 65L05 · 65L06 · 65L07 · 65L020

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Time‐adaptive multirate generalized‐structure additively partitioned Runge‐Kutta schemes for mechanical fluid‐structure interaction

Bremicker-Trübelhorn

Ortleb

2016

Proc Appl Math and Mech

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We extend the multirate GARK approach by Günther and Sandu [1, 2] to the case of non‐constant micro steps per macro step to be able to cover time adaptivity, arriving at our so‐called TAMGARK scheme and derive order conditions for arbitrary non‐constant micro step sizes. The TAMGARK scheme is applied to the classical piston problem [3] to study the interaction between the piston and inviscid fluid flow in one spatial dimension. We demonstrate the functionality of the chosen time‐adaptation strategy and the ability of the overall scheme to preserve the order of the basic schemes. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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A Generalized-Structure Approach to Additive Runge--Kutta Methods

Cited by 63 publications

References 19 publications

On Multirate GARK Schemes with Adaptive Micro Step Sizes for Fluid–Structure Interaction: Order Conditions and Preservation of the Geometric Conservation Law

On Multirate GARK Schemes with Adaptive Micro Step Sizes for Fluid–Structure Interaction: Order Conditions and Preservation of the Geometric Conservation Law

Multirate generalized additive Runge Kutta methods

Time‐adaptive multirate generalized‐structure additively partitioned Runge‐Kutta schemes for mechanical fluid‐structure interaction

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