Improving the Laplace transform integration method

Lynch, Peter; Clancy, Colm

doi:10.1002/qj.2670

Cited by 5 publications

(7 citation statements)

References 16 publications

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“…In particular, its low phase errors in the triadic test cases are a reflection of its exact representation of the linear terms. As well as any exponential time-differencing scheme that has no linear, dispersive errors, other methods such as a Laplace integrator, which can use a larger time-step of a semi-implicit method (Lynch & Clancy, 2016), are of interest. Future work will consider other time-steppers that can use the accuracy of the explicit methods beyond the linear stability limits.…”

Section: Discussionmentioning

confidence: 99%

Quantifying the accuracy of large time-steps in highly oscillatory systems

Andrews¹,

Shipton²

2023

Preprint

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<p>There is an ever-increasing demand for longer and higher resolution numerical simulations of the weather and climate. To achieve this with reasonable wall-clock times, it is desirable to use as large a time-step as possible, whilst retaining a stable and accurate solution. However, this is very challenging in the presence of highly oscillatory linear waves; there are explicit time-step limits and losses in accuracy with implicit methods. This talk will highlight where non-linear errors can result from large time-steps and provide metrics for quantifying this. We begin by re-casting the non-linearity as a product of linear waves. In the Rotating Shallow Water Equations (RSWEs), this allows for key dynamics to be expressed as three-wave &#8216;triad&#8217; interactions. A non-linear &#8216;triadic&#8217; time-stepping error is computed using linear stability polynomials. A number of explicit and implicit time-stepping methods (such as RK4, TR-BDF2, ETD-RK2) will be compared analytically in the RSWEs. Next, two new test problems enable analyses of large time-step simulations. The first is of a Gaussian perturbation to a RSWE height field. A proposed metric, relating to the kinetic energy distribution over temporal frequency, quantifies phase errors in the height reformation. The second test case initialises linear waves which, via direct- and near- resonant triad interactions, will construct non-linear dynamics. Phase errors with large time-steps can be identified in the corresponding height fields. A first variant of this case will initialise only two waves; this will primarily instigate an energy exchange within a dominant triad. A second version, containing more slow modes, enables a re-distribution of fast mode energy into rings in wavenumber space.</p>

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Section: Discussionmentioning

confidence: 99%

Quantifying the accuracy of large time-steps in highly oscillatory systems

Andrews¹,

Shipton²

2023

Preprint

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“…The LT scheme filters high frequency components by using a modified inversion operator L * : this can be done numerically by distorting the Bromwich contour for the inversion integral to a closed curve excluding poles associated with the high frequencies, as in [2,3,17,18,28]. In the persent study, as in those of Lynch and Clancy [20] and Harney and Lynch [12], we invert the transform analytically, explicitly eliminating components with frequency greater than a specified cut-off frequency ω c .…”

Section: Filtering With the Lt Schemementioning

confidence: 99%

“…It was pointed out in Lynch & Clancy [20] that the Laplace transform (LT) method with analytic inversion gives an exact treatment of the linear modes. This is due to the fact that the LT scheme does not involve time-averaging of the linear terms.…”

Section: Introductionmentioning

confidence: 99%

A Lagrange-Laplace Integration Scheme for Weather Prediction and Climate Modelling

Lynch¹

2022

Preprint

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A time integration scheme based on semi-Lagrangian advection and Laplace transform adjustment has been implemented in a baroclinic primitive equation model. The semi-Lagrangian scheme makes it possible to use large time steps. However, errors arising from the semi-implicit scheme increase with the time step size. In contrast, the errors using the Laplace transform adjustment remain relatively small for typical time steps used with semi-Lagrangian advection. Numerical experiments confirm the superior performance of the Laplace transform scheme relative to the semi-implicit reference model. The algorithmic complexity of the scheme is comparable to the reference model, making it computationally competitive, and indicating its potential for integrating weather and climate prediction models.

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“…In this article, as in Lynch and Clancy (), we invert the transform analytically, explicitly eliminating components with frequency greater than a specified cut‐off frequency ω c . This may be done with a sharp cut‐off at ω c , or with a smooth function such as a Butterworth filter having frequency response function

scriptH false(ω false) = \frac{1}{1 + {false(ω false/ ω_{normalc} false)}^{L}} .

…”

Section: Filtering With the Lt Schemementioning

confidence: 99%

“…All the above schemes used a numerical inversion of the Laplace transform. Lynch and Clancy (2016) showed that, with an appropriate formulation of the equations, the inversion could be performed analytically. This greatly enhanced the computational efficiency of the LT scheme, making it competitive with the SI method.…”

Section: Introductionmentioning

confidence: 99%

Laplace transform integration of a baroclinic model

Harney

Lynch

2019

Quart J Royal Meteoro Soc

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A time integration scheme based on the Laplace transform (LT) has been implemented in a baroclinic primitive equation model. The LT scheme provides an attractive alternative to the popular semi-implicit (SI) scheme. Analysis shows that it is more accurate than SI for both linear and nonlinear terms of the equations. Numerical experiments confirm the superior performance of the LT scheme. The algorithmic complexity of the LT scheme is comparable with SI, with just an additional transformation to vertical eigenmodes each time step, and it provides the possibility of improving weather forecasts at comparable computational cost. KEYWORDS filtering, forecast accuracy, Laplace transform, numerical weather prediction, time integration 1 Q J R Meteorol Soc. 2019;145:347-355.wileyonlinelibrary.com/journal/qj

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Improving the Laplace transform integration method

Cited by 5 publications

References 16 publications

Quantifying the accuracy of large time-steps in highly oscillatory systems

Quantifying the accuracy of large time-steps in highly oscillatory systems

A Lagrange-Laplace Integration Scheme for Weather Prediction and Climate Modelling

Laplace transform integration of a baroclinic model

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