A Linearly Fourth Order Multirate Runge–Kutta Method with Error Control

Fok, Pak-Wing

doi:10.1007/s10915-015-0017-4

Cited by 13 publications

(10 citation statements)

References 18 publications

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“…Adaptive time discretization methods are the state of the art for ODE solvers, while, for PDE solvers, spatial adaptivity techniques are also widely used. Local time step adaptation is feasible in the framework of so-called local time stepping or multirate approaches, where different components of the system can have different time step sizes (see Bonaventura et al, 2020; Carciopolo et al, 2019; Fok, 2015; Gear and Wells, 1984; Sandu, 2019; Savcenco et al, 2007; Seny et al, 2014), which are however still far from mainstream for most applications. For PDE solvers, local spatial adaptivity techniques are also very common (Bangerth and Rannacher, 2013; Bangerth et al, 2012), but their incorporation in operational applications is still a research topic (see, for example,.…”

Section: Future Directionsmentioning

confidence: 99%

Resiliency in numerical algorithm design for extreme scale simulations

Agullo

Altenbernd

Anzt

et al. 2021

The International Journal of High Performance Computing Applica

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This work is based on the seminar titled ‘Resiliency in Numerical Algorithm Design for Extreme Scale Simulations’ held March 1–6, 2020, at Schloss Dagstuhl, that was attended by all the authors. Advanced supercomputing is characterized by very high computation speeds at the cost of involving an enormous amount of resources and costs. A typical large-scale computation running for 48 h on a system consuming 20 MW, as predicted for exascale systems, would consume a million kWh, corresponding to about 100k Euro in energy cost for executing 1023 floating-point operations. It is clearly unacceptable to lose the whole computation if any of the several million parallel processes fails during the execution. Moreover, if a single operation suffers from a bit-flip error, should the whole computation be declared invalid? What about the notion of reproducibility itself: should this core paradigm of science be revised and refined for results that are obtained by large-scale simulation? Naive versions of conventional resilience techniques will not scale to the exascale regime: with a main memory footprint of tens of Petabytes, synchronously writing checkpoint data all the way to background storage at frequent intervals will create intolerable overheads in runtime and energy consumption. Forecasts show that the mean time between failures could be lower than the time to recover from such a checkpoint, so that large calculations at scale might not make any progress if robust alternatives are not investigated. More advanced resilience techniques must be devised. The key may lie in exploiting both advanced system features as well as specific application knowledge. Research will face two essential questions: (1) what are the reliability requirements for a particular computation and (2) how do we best design the algorithms and software to meet these requirements? While the analysis of use cases can help understand the particular reliability requirements, the construction of remedies is currently wide open. One avenue would be to refine and improve on system- or application-level checkpointing and rollback strategies in the case an error is detected. Developers might use fault notification interfaces and flexible runtime systems to respond to node failures in an application-dependent fashion. Novel numerical algorithms or more stochastic computational approaches may be required to meet accuracy requirements in the face of undetectable soft errors. These ideas constituted an essential topic of the seminar. The goal of this Dagstuhl Seminar was to bring together a diverse group of scientists with expertise in exascale computing to discuss novel ways to make applications resilient against detected and undetected faults. In particular, participants explored the role that algorithms and applications play in the holistic approach needed to tackle this challenge. This article gathers a broad range of perspectives on the role of algorithms, applications and systems in achieving resilience for extreme scale simulations. The ultimate goal is to spark novel ideas and encourage the development of concrete solutions for achieving such resilience holistically.

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Section: Future Directionsmentioning

confidence: 99%

Resiliency in numerical algorithm design for extreme scale simulations

Agullo

Altenbernd

Anzt

et al. 2021

The International Journal of High Performance Computing Applica

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“…We now describe in detail the time step refinement and partitioning strategy that we have used in the multirate algorithm described in section 2. Our approach is based on the strategy proposed for an explicit Runge Kutta multirate method in [7], where the time steps for refinement are obtained from the error estimates of the global step. The user specified tolerance plays an important role in the partitioning of the system.…”

Section: The Time Step Refinement and Partitioning Criterionmentioning

confidence: 99%

“…In particular, the cubic Hermite interpolant can be employed to maintain higher accuracy also for stiff problems in the framework of a multirate approach. The proposed self-adjusting multirate TR-BDF2 method is equipped here with a partitioning and time step selection criterion based on the technique proposed in [7].…”

Section: Introductionmentioning

confidence: 99%

A self adjusting multirate algorithm based on the TR-BDF2 method

Bonaventura,

Casella,

Delpopolo

et al. 2018

Preprint

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We propose a self adjusting multirate method based on the TR-BDF2 solver. The potential advantages of using TR-BDF2 as the key component of a multirate framework are highlighted. A linear stability analysis of the resulting approach is presented and the stability features of the resulting algorithm are analysed. The analysis framework is completely general and allows to study along the same lines the stability of self adjusting multirate methods based on a generic one step solver. A number of numerical experiments demonstrate the efficiency and accuracy of the resulting approach also the time discretization of hyperbolic partial differential equations.

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“…More recently, many methods have been investigated as starting points for more sophisticated LTS methods, including both substep [5,6,7,8,9,10] and multistep [2,11,12] integrators and also less common methods such as leapfrog [11,13], Richardson extrapolation [14], ADER [15], and implicit methods [16]. Demirel et al [17] have even explored LTS schemes constructed from multiple unrelated GTS integrators.…”

Section: Introductionmentioning

confidence: 99%

A High-Order, Conservative Integrator with Local Time-Stepping

Throwe¹,

Teukolsky²

2020

SIAM J. Sci. Comput.

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We present a family of multistep integrators based on the Adams-Bashforth methods. These schemes can be constructed for arbitrary convergence order with arbitrary step size variation. The step size can differ between different subdomains of the system. It can also change with time within a given subdomain. The methods are linearly conservative, preserving a wide class of analytically constant quantities to numerical roundoff, even when numerical truncation error is significantly higher. These methods are intended for use in solving conservative PDEs in discontinuous Galerkin formulations, but are applicable to any system of ODEs. A numerical test demonstrates these properties and shows that significant speed improvements over the standard Adams-Bashforth schemes can be obtained.

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A Linearly Fourth Order Multirate Runge–Kutta Method with Error Control

Cited by 13 publications

References 18 publications

Resiliency in numerical algorithm design for extreme scale simulations

Resiliency in numerical algorithm design for extreme scale simulations

A self adjusting multirate algorithm based on the TR-BDF2 method

A High-Order, Conservative Integrator with Local Time-Stepping

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