Multirate generalized additive Runge Kutta methods

Günther, Michael; Sandu, Adrian

doi:10.1007/s00211-015-0756-z

Cited by 70 publications

(86 citation statements)

References 39 publications

(75 reference statements)

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“…A drawback of these aforementioned early schemes is the fact that the coupling between slow and fast components of the system is done by interpolating and extrapolating state variables. This complicates the implementation into existing simulation packages [18]. Kvaernø and Rentrop overcame this shortcoming by developing a concise theory of explicit multirate Runge-Kutta schemes and obtained promising results for electrical network simulations [19].…”

Section: Introductionmentioning

confidence: 99%

“…Deriving information by interpolation and extrapolation was circumvented by using the internal Runge-Kutta stages for coupling the integration schemes for the subsystems. In [18], a generalization to stiff problems was proposed. In that work, explicit Runge-Kutta schemes integrating the (nonstiff) fast system component are combined with linearly implicit Rosenbrock-Wanner schemes integrating the (stiff) slow system component.…”

Section: Introductionmentioning

confidence: 99%

“…The GARK framework gives the intuition necessary to define new classes of additive schemes and eases the construction of additive schemes which preserve the order of the base ones. Günther and Sandu made use of the GARK framework to derive a multirate scheme solely based on implicit and explicit Runge-Kutta schemes [26]. Günther and Sandu's multirate GARK scheme carries out just one so-called macro step of a "large" step size H within the slow component per iteration while at the same time carrying out several so-called micro steps of a "small" step size h < H within the fast component per iteration, until both components "meet up" again.…”

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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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

View full text Add to dashboard Cite

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“…General expressions for the local errors, that can be used to derive error bounds when both Δt and Δx tend to zero, are given in Sect. 4. Detailed error bounds are found in Sect.…”

Section: Introductionmentioning

confidence: 99%

Error Analysis of Explicit Partitioned Runge–Kutta Schemes for Conservation Laws

2014

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An error analysis is presented for explicit partitioned Runge-Kutta methods and multirate methods applied to conservation laws. The interfaces, across which different methods or time steps are used, lead to order reduction of the schemes. Along with cell-based decompositions, also flux-based decompositions are studied. In the latter case mass conservation is guaranteed, but it will be seen that the accuracy may deteriorate.

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A volume‐agglomeration multirate time advancing for high Reynolds number flow simulation

Itam

Wornom

Koobus

et al. 2018

Numerical Methods in Fluids

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Summary A frequent configuration in computational fluid mechanics combines an explicit time advancing scheme for accuracy purposes and a computational grid with a very small portion of much smaller elements than in the remaining mesh. Two examples of such situations are the travel of a discontinuity followed by a moving mesh, and the large eddy simulation of high Reynolds number flows around bluff bodies where together very thin boundary layers and vortices of much more important size need to be captured. For such configurations, multistage explicit time advancing schemes with global time stepping are very accurate but very CPU consuming. In order to reduce this problem, the multirate (MR) time stepping approach represents an interesting improvement. The objective of such schemes, which allow to use different time steps in the computational domain, is to avoid penalizing the computational cost of the time advancement of unsteady solutions that would become large due to the use of small global time steps imposed by the smallest elements such as those constituting the boundary layers. In the present work, a new MR scheme based on control volume agglomeration is proposed for the solution of the compressible Navier‐Stokes equations equipped with turbulence models. The method relies on a prediction step where large time steps are performed with an evaluation of the fluxes on macrocells for the smaller elements for stability purpose and a correction step in which small time steps are employed. The accuracy and efficiency of the proposed method are evaluated on several benchmarks flows: the problem of a moving contact discontinuity (inviscid flow), the computation with a hybrid turbulence model of flows around bluff bodies like a flow around a space probe model at Reynolds number 106, a circular cylinder at Reynolds number 8.4 × 106, and two tandem cylinders at Reynolds number 1.66 × 105 and 1.4 × 105.

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Multirate generalized additive Runge Kutta methods

Cited by 70 publications

References 39 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

Error Analysis of Explicit Partitioned Runge–Kutta Schemes for Conservation Laws

A volume‐agglomeration multirate time advancing for high Reynolds number flow simulation

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