Exponential integrators

Hochbruck, Marlis; Ostermann, Alexander

doi:10.1017/s0962492910000048

Cited by 983 publications

(943 citation statements)

References 128 publications

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“…Promising approaches for satisfying the latter condition are exponential time integrators [36,47]; (b) to overcome the overly restrictive time-step limitations of EBTI schemes combined with highly scalable horizontal discretizations, either through horizontal/vertical splitting (HEVI) [2,8,40] or through combining SISL PBTI methods with discontinuous Galerkin (DG) discretization [99]; and (c) to further the scalability and the adaptation of algorithms to emerging HPC architectures involving SE [32] or fully-implicit time-stepping approaches [113], and further through exploiting additional parallelism with time-parallel algorithms [33].…”

Section: Discussion and Concluding Remarksmentioning

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: Discussion and Concluding Remarksmentioning

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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“…In that case it makes sense to couple the ideas of this paper with the use of different versions of operatorial splitting, e.g. the Strang splitting (Iserles 2008), exponential integrators (Hochbruck & Ostermann 2010) or IMEX-type methods (Kassam & Trefethen 2005). Consider, for example, the reaction-diffusion equation…”

Section: Skew-symmetry and Time-stepping Algorithmsmentioning

confidence: 99%

“…It is clear, however, that space and time should not be discretised in isolation. We must fashion time-stepping methods not just by adopting known ODE solvers but by developing bespoke methods to cope with specific PDEs, like in (Hochbruck & Ostermann 2010) and (Shu & Osher 1988). Likewise -and this is the main message of this paper and, indeed, of the entire history of numerical stability -we need often to rethink our space discretisation to render time-stepping stable.…”

Section: Skew-symmetry and Time-stepping Algorithmsmentioning

confidence: 99%

Numerical Stability in the Presence of Variable Coefficients

Hairer

Iserles

2015

Found Comput Math

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The main concern of this paper is with the stable discretisation of linear partial differential equations of evolution with time-varying coefficients. We commence by demonstrating that an approximation of the first derivative by a skew-symmetric matrix is fundamental in ensuring stability for many differential equations of evolution. This motivates our detailed study of skew-symmetric differentiation matrices for univariate finite-difference methods.We prove that, in order to sustain a skew-symmetric differentiation matrix of order p ≥ 2, a grid must satisfy 2p − 3 polynomial conditions. Moreover, once it satisfies these conditions, it supports a banded skew-symmetric differentiation matrix of this order and of the bandwidth 2p − 1 which can be derived in a constructive manner.Some applications require not just skew-symmetry but also that the growth in the elements of the differentiation matrix is at most linear in the number of unknowns. This is always true for our tridiagonal matrices of order 2 but need not be true otherwise, a subject which we explore further.Another subject which we examine is the existence and practical construction of grids that support skew-symmetric differentiation matrices of a given order. We resolve this issue completely for order-two methods.We conclude the paper with a list of open problems and their discussion.0 Communicated by Peter Olver 0

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“…The inhomogeneous vector f n consists of boundary contributions (Hochbruck & Ostermann 2010, Iserles 1992) and we note that f n ≡ 0 when the Dirichlet boundary conditions are zero. Since e 1 2 (∆t)C e (∆t)D2 e 1 2 (∆t)C = e (∆t)(D2+C) + O (∆t) 3 , the method is second order in time, therefore it makes sense to discretise in space to a similar order -this is the case with the familiar differentiation matrices We do not consider in this paper the practicalities of computing the exponentials in (1.2), whether exactly or by some sort of approximation.…”

Section: Introductionmentioning

confidence: 99%

“…However, the universality of the Sheng barrier has been recently challenged from three different quarters. Firstly, a new breed of ODE solvers, exponential integrators, have been methodically applied to semidiscretised PDEs, allowing for the use of high-order time discretisation, combined with the use of exponential functions (Hochbruck & Ostermann 2010). Secondly, (Hansen & Ostermann 2009) have demonstrated that using complex-valued time it is possible to breach the Sheng barrier.…”

Section: Introductionmentioning

confidence: 99%

On skew-symmetric differentiation matrices

Iserles

2013

IMA Journal of Numerical Analysis

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The theme of this paper is the construction of finite-difference approximations to the first derivative in the presence of Dirichlet boundary conditions. Stable implementation of splitting-based discretisation methods for the convectiondiffusion equation requires the underlying matrix to be skew symmetric and this turns out to be a surprisingly restrictive condition. We prove that no skewsymmetric approximation on an equidistant grid may exceed order two. Once non-equidistant grid is allowed, this barrier can be breached.

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Exponential integrators

Cited by 983 publications

References 128 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

Numerical Stability in the Presence of Variable Coefficients

On skew-symmetric differentiation matrices

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