Multirate GARK Schemes for Multiphysics Problems

Günther, Michael; Hachtel, Christoph; Sandu, Adrian

doi:10.1007/978-3-319-30399-4_12

Cited by 3 publications

(8 citation statements)

References 2 publications

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“…using Equations (24) and (27). The corresponding condition on the right hand side is then fulfilled due to Equation (26).…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…This assumption is incorporated into the Equations (24), (27), (31), (33), respectively, depending on the considered order, as well as the Equation (26). Thus, the MGARK order conditions of Theorem 2 are sufficient, but not necessary.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…An internally consistent MGARK scheme as specified by Definitions 4 and 5 is at least of first order if Equations (24)- (26) hold. The scheme is at least of second order, if in addition to the first order conditions the Equations (27) and (28) hold. The scheme is at least of third order, if in addition to the second order conditions the third order conditions for the individual schemes, i.e., Equations (31)- (34) as well as the coupling Equations (35) and (36) are fulfilled.…”

Section: Definition 5 a Gark(d D J) Scheme Given By Definition 2 Imentioning

confidence: 99%

“…The GARK formalism includes a variety of well-known schemes for partitioned problems, like the classical implicit-explicit (IMEX) Runge-Kutta schemes [20,23], as well as former multirate methods as in [19,21]. Furthermore, Günther, Hachtel and Sandu showed the feasibility of the multirate GARK approach for both implicit-implicit and implicit-explicit pairing of basic schemes [27] in the context of thermal-electrical coupling.…”

Section: Introductionmentioning

confidence: 99%

“…In contrast to the original application from electrical circuit simulation [18,19] and the test examples from thermal-electrical coupling [27], the fluid is expected to be stiff in many applications, which is the reason why implicit schemes often need to be applied to the fast fluid part. In case of (partially) explicit time integration of the fluid equations, e.g., in advection-dominated flow, the time steps may be requested to change gradually as a consequence of changes of the characteristic speeds.…”

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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“…using Equations (24) and (27). The corresponding condition on the right hand side is then fulfilled due to Equation (26).…”

Section: Proof Of Theoremmentioning

confidence: 99%

Section: Proof Of Theoremmentioning

confidence: 99%

Section: Definition 5 a Gark(d D J) Scheme Given By Definition 2 Imentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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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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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Design of High-Order Decoupled Multirate GARK Schemes

Sarshar¹,

Roberts²,

Sandu³

2019

SIAM J. Sci. Comput.

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Multirate time integration methods apply different step sizes to resolve different components of the system based on the local activity levels. This local selection of step sizes allows increased computational efficiency while achieving the desired solution accuracy. 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.Seeking to overcome these challenges, this work focuses on the design of practical high-order multirate methods using the theoretical framework of generalized additive Runge-Kutta (MrGARK) methods [19], which provides the generic order conditions and the linear and nonlinear stability analyses. A set of design criteria for practical multirate methods is defined herein: method coefficients should be generic in the step size ratio, but should not depend strongly on this ratio; unnecessary coupling between the fast and the slow components should be avoided; and the step size controllers should adjust both the micro-and the macro-steps. Using these criteria, we develop MrGARK schemes of up to order four that are explicit-explicit (both the fast and slow component are treated explicitly), implicit-explicit (implicit in the fast component and explicit in the slow one), and explicit-implicit (explicit in the fast component and implicit in the slow one). Numerical experiments illustrate the performance of these new schemes.

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Multirate GARK Schemes for Multiphysics Problems

Cited by 3 publications

References 2 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

Design of High-Order Decoupled Multirate GARK Schemes

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