The effect of linear dispersive errors on nonlinear time-stepping accuracy

For simulations of time-evolution problems, such as weather and climate models, taking the largest stable time-step is advantageous for reducing the wall-clock time. We propose methods for studying the effect of linear dispersive errors on the time-stepping accuracy of nonlinear problems. We demonstrate an application of this to the Rotating Shallow Water Equations (RSWEs). To begin, a nonlinear time-stepping `triadic error' metric is constructed from three-wave interactions. Stability polynomials, obtained from the oscillatory Dahlquist test equation, enable the computation of triadic errors for different time-steppers; we compare five classical schemes. We next provide test cases comparing different time-step sizes within a numerical model. The first case is of a reforming Gaussian height perturbation. This contains a nonlinear phase shift that can be missed with a large time-step. The second set of test cases initialise individual waves to allow specific triads to form. The presence of a slow transition from linear to nonlinear dynamics creates a good venue for testing how the slow phase information is replicated with a large time-step. Three models, including the finite element code Gusto, and the MetOffice's new LFRic model, are examined in these test cases with different time-steppers.

[1]  C. Garfinkel,et al.  Tropical background and wave spectra: contribution of wave-wave interactions in a moderately nonlinear turbulent flow , 2021, Journal of the Atmospheric Sciences.

[2]  C. Cotter,et al.  Time parallel integration and phase averaging for the nonlinear shallow water equations on the sphere , 2021, ArXiv.

[3]  C. Cotter,et al.  Higher order phase averaging for highly oscillatory systems , 2021, Multiscale Model. Simul..

[4]  A. Owen.  Resonant effects in weakly nonlinear geophysical fluid dynamics , 2019 .

[5]  F. Giraldo,et al.  Current and Emerging Time-Integration Strategies in Global Numerical Weather and Climate Prediction , 2019 .

[6]  Colin J. Cotter,et al.  A mixed finite‐element, finite‐volume, semi‐implicit discretization for atmospheric dynamics: Cartesian geometry , 2019, Quarterly Journal of the Royal Meteorological Society.

[7]  N. Paldor,et al.  The Matsuno baroclinic wave test case , 2018, Geoscientific Model Development.

[8]  Vu Thai Luan,et al.  Further development of efficient and accurate time integration schemes for meteorological models , 2018, J. Comput. Phys..

[9]  A. Peddle,et al.  Components of nonlinear oscillation and optimal averaging for stiff PDEs , 2018 .

[10]  R. Heikes,et al.  DCMIP2016: A Review of Non-hydrostatic Dynamical Core Design and Intercomparison of Participating Models , 2017 .

[11]  G. Flierl,et al.  Wavenumber-frequency analysis of single-layer shallow-water beta-plane quasi-geostrophic turbulence , 2017 .

[12]  Sophie Valcke,et al.  Crossing the chasm: how to develop weather and climate models for next generation computers? , 2017 .

[13]  N. Paldor,et al.  A quantitative test case for global‐scale dynamical cores based on analytic wave solutions of the shallow‐water equations , 2016 .

[14]  Per-Gunnar Martinsson,et al.  A high-order time-parallel scheme for solving wave propagation problems via the direct construction of an approximate time-evolution operator , 2016 .

[15]  Mohamed Zerroukat,et al.  A moist Boussinesq shallow water equations set for testing atmospheric models , 2015, J. Comput. Phys..

[16]  P. Lauritzen,et al.  A high‐order fully explicit flux‐form semi‐Lagrangian shallow‐water model , 2014 .

[17]  Matthias Aechtner,et al.  A conservative adaptive wavelet method for the shallow‐water equations on the sphere , 2013, 1404.0405.

[18]  Terry Haut,et al.  An Asymptotic Parallel-in-Time Method for Highly Oscillatory PDEs , 2013, SIAM J. Sci. Comput..

[19]  Jianping Li,et al.  Space–Time Spectral Analysis of the Southern Hemisphere Daily 500-hPa Geopotential Height , 2012 .

[20]  Colin J. Cotter,et al.  Computational Modes and Grid Imprinting on Five Quasi-Uniform Spherical C Grids , 2012 .

[21]  Colin J. Cotter,et al.  Mixed finite elements for numerical weather prediction , 2011, J. Comput. Phys..

[22]  W. Dewar,et al.  Scattering of gravity waves by potential vorticity in a shallow-water fluid , 2010, Journal of Fluid Mechanics.

[23]  Bram van Leer,et al.  High-order finite-volume methods for the shallow-water equations on the sphere , 2010, J. Comput. Phys..

[24]  M. Hochbruck,et al.  Exponential integrators , 2010, Acta Numerica.

[25]  Sohan Dharmaraja,et al.  An analysis of the TR-BDF2 integration scheme , 2009 .

[26]  G. Vallis.  Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation , 2017 .

[27]  Leslie M. Smith,et al.  On near resonances and symmetry breaking in forced rotating flows at moderate Rossby number , 2005, Journal of Fluid Mechanics.

[28]  R. K. Scott,et al.  An initial-value problem for testing numerical models of the global shallow-water equations , 2004 .

[29]  S. Cox,et al.  Exponential Time Differencing for Stiff Systems , 2002 .

[30]  Leslie M. Smith,et al.  Transfer of energy to two-dimensional large scales in forced, rotating three-dimensional turbulence , 1999 .

[31]  Matthew C. Wheeler,et al.  Convectively Coupled Equatorial Waves: Analysis of Clouds and Temperature in the Wavenumber–Frequency Domain , 1999 .

[32]  Andrew J. Majda,et al.  Low Froude number limiting dynamics for stably stratified flow with small or finite Rossby numbers , 1998 .

[33]  Andrew J. Majda,et al.  Averaging over fast gravity waves for geophysical flows with arbitrary potential vorticity , 1996 .

[34]  Lawrence F. Shampine,et al.  Analysis and implementation of TR-BDF2 , 1996 .

[35]  S. Schochet.  Fast Singular Limits of Hyperbolic PDEs , 1994 .

[36]  P. Swarztrauber,et al.  A standard test set for numerical approximations to the shallow water equations in spherical geometry , 1992 .

[37]  J. Boyd.  Chebyshev and Fourier Spectral Methods , 1989 .

[38]  F. Verhulst,et al.  Averaging Methods in Nonlinear Dynamical Systems , 1985 .

[39]  Cleve B. Moler,et al.  Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later , 1978, SIAM Rev..

[40]  Y. Hayashi,et al.  Space-Time Spectral Analysis of Mid-Latitude Disturbances Appearing in a GFDL General Circulation Model. , 1977 .

[41]  A. Newell.  Rossby wave packet interactions , 1969, Journal of Fluid Mechanics.

[42]  David A. Pope.  An exponential method of numerical integration of ordinary differential equations , 1963, CACM.

[43]  P. Lynch,et al.  Improving the Laplace transform integration method , 2016 .

[44]  John Thuburn,et al.  Horizontal grids for global weather and climate prediction models: a review , 2012 .

[45]  A. Iserles.  A First Course in the Numerical Analysis of Differential Equations: Gaussian elimination for sparse linear equations , 2008 .

[46]  A. Majda.  Introduction to PDEs and Waves in Atmosphere and Ocean , 2003 .