On the Construction and Comparison of Difference Schemes

Strang, Gilbert

doi:10.1137/0705041

Cited by 3,087 publications

(2,000 citation statements)

References 8 publications

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“…The analyses focus on the accuracy and stability implied for components of simple linear systems (acoustic and gravity waves, and advection); and the performance of numerical implementations for idealised (dry) atmospheric tests, often compared to solutions from a very high-resolution highorder explicit RK method. Substantially different approaches have been recommended: some completely splitting the vertical and horizontal integrations using Strang-type splitting [8,97,100]; others proposing schemes that keep the vertical and horizontal solutions balanced in time, by integrating over the same time-interval, at each predictorstage [40,63,106]. In keeping with the semi-implicit approach (Sect.…”

Section: Horizontally-explicit Vertically-implicit (Hevi) Schemesmentioning

confidence: 99%

Current and Emerging Time-Integration Strategies in Global Numerical Weather and Climate Prediction

Mengaldo

Wyszogrodzki

Diamantakis

et al. 2018

Arch Computat Methods Eng

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The continuous partial differential equations governing a given physical phenomenon, such as the Navier-Stokes equations describing the fluid motion, must be numerically discretized in space and time in order to obtain a solution otherwise not readily available in closed (i.e., analytic) form. While the overall numerical discretization plays an essential role in the algorithmic efficiency and physically-faithful representation of the solution, the time-integration strategy commonly is one of the main drivers in terms of cost-to-solution (e.g., time-or energy-to-solution), accuracy and numerical stability, thus constituting one of the key building blocks of the computational model. This is especially true in time-critical applications, including numerical weather prediction (NWP), climate simulations and engineering. This review provides a comprehensive overview of the existing and emerging time-integration (also referred to as time-stepping) practices used in the operational global NWP and climate industry, where global refers to weather and climate simulations performed on the entire globe. While there are many flavors of time-integration strategies, in this review we focus on the most widely adopted in NWP and climate centers and we emphasize the reasons why such numerical solutions were adopted. This allows us to make some considerations on future trends in the field such as the need to balance accuracy in time with substantially enhanced time-to-solution and associated implications on energy consumption and running costs. In addition, the potential for the co-design of time-stepping algorithms and underlying high performance computing hardware, a keystone to accelerate the computational performance of future NWP and climate services, is also discussed in the context of the demanding operational requirements of the weather and climate industry.

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Section: Horizontally-explicit Vertically-implicit (Hevi) Schemesmentioning

confidence: 99%

Current and Emerging Time-Integration Strategies in Global Numerical Weather and Climate Prediction

Mengaldo

Wyszogrodzki

Diamantakis

et al. 2018

Arch Computat Methods Eng

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“…where γ m k is defined as in (15). This update corresponds to (16) for the onedimensional shallow water equations and (25) for the two-dimensional scalar advection equation.…”

Section: D Systems Of Equations: Application To Shallow Water Equationsmentioning

confidence: 99%

A 2D extension of a Large Time Step explicit scheme (CFL>1) for unsteady problems with wet/dry boundaries

Morales-Hernändez

Hubbard

García‐Navarro

2014

Journal of Computational Physics

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A 2D Large TimeStep (LTS) explicit scheme on structured grids is presented in this work. It is first detailed and analysed for the 2D linear advection equation and then applied to the 2D shallow water equations. The dimensional splitting technique allows us to extend the ideas developed in the 1D case related to source terms, boundary conditions and the reduction of the time step in the presence of large discontinuities. The boundary conditions treatment as well as the wet/dry fronts in the case of the 2D shallow water equations require extra effort. The proposed scheme is tested on linear and non-linear equations and systems, with and without source terms. The numerical results are compared with those of the conventional scheme as well as with analytical solutions and experimental data.

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“…These dedicated methods chosen for each subsystem are then responsible for dealing with the fast scales associated with each one of them, in a separate manner, while the reconstruction of the global solution by the splitting scheme should guarantee an accurate description with error control of the global physical coupling, without being related to the stability constraints of the numerical resolution of each subsystem. A second order Strang scheme is then implemented [12] …”

Section: Adaptive Time Operator Splittingmentioning

confidence: 99%

Adaptive Time-Space Algorithms for the Simulation of Multi-scale Reaction Waves

Duarte

Massot

Descombes³

et al. 2011

Finite Volumes for Complex Applications VI Problems &Amp; Perspectives

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We present a new resolution strategy for multi-scale reaction waves based on adaptive time operator splitting and space adaptive multiresolution, in the context of localized and stiff reaction fronts. The main goal is to perform computationally efficient simulations of the dynamics of multi-scale phenomena under study, considering large simulation domains with conventional computing resources. We aim at time-space accuracy control of the solution and splitting time steps purely dictated by the physics of the phenomenon and not by stability constraints associated with mesh size or source time scales. Numerical illustrations are provided for 2D and 3D combustion applications modeled by reaction-convection-diffusion equations.

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On the Construction and Comparison of Difference Schemes

Cited by 3,087 publications

References 8 publications

Current and Emerging Time-Integration Strategies in Global Numerical Weather and Climate Prediction

Current and Emerging Time-Integration Strategies in Global Numerical Weather and Climate Prediction

A 2D extension of a Large Time Step explicit scheme (CFL>1) for unsteady problems with wet/dry boundaries

Adaptive Time-Space Algorithms for the Simulation of Multi-scale Reaction Waves

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