• Bates, J. R., and A. McDonald, 1982: Multiply upstream, semi-Lagrangian advective scheme: Analysis and application to a multilevel primitive equation model. Mon. Wea. Rev.,110, 1831–1842.

  • Burridge, D. M., 1975: A split semi-implicit reformulation of the Bushby-Timpson 10 level model. Quart. J. Roy. Meteor. Soc.,101, 777–792.

  • Gadd, A. J., 1978: A split explicit integration scheme. Quart. J. Roy. Meteor. Soc.,104, 569–582.

  • Gear, C. W., 1971: Numerical Initial Value Problems in Ordinary Differential Equations. Prentice Hall, 253 pp.

  • Goldstein, H., 1980: Classical Mechanics. Addison Wesley, 672 pp.

  • Janjić, Z. I., 1995: A note on the performance of the multiply-upstream semi-Lagrangian advection schemes for one-dimensional nonlinear momentum conservation equation. Meteor. Atmos. Phys.,55, 1–16.

  • Leslie, L. M., and R. J. Purser, 1991: High-order numerics in an unstaggered three-dimensional time-split semi-Lagrangian forecast model. Mon. Wea. Rev.,119, 1612–1623.

  • ——, and ——, 1995: Three-dimensional mass-conserving semi-Lagrangian scheme employing forward trajectories. Mon. Wea. Rev.,123, 2551–2566.

  • Lorenz, E. N., 1971: An N-cycle time-differencing scheme for stepwise numerical integration. Mon. Wea. Rev.,99, 644–648.

  • McDonald, A., 1986: A semi-Lagrangian and semi-implicit two-time-level integration scheme. Mon. Wea. Rev.,114, 824–830.

  • McGregor, J. L., 1987: Accuracy and initialization of a two-time-level semi-Lagrangian model. Short- and Medium-Range Numerical Weather Prediction, T. Matsuno, Ed., Special Vol. of the J. Meteor. Soc. Japan, 233–246.

  • ——, 1993: Economical determination of departure points for semi-Lagrangian models. Mon. Wea. Rev.,121, 221–230.

  • Purser, R. J., and L. M. Leslie, 1988: A semi-implicit semi-Lagrangian finite-difference scheme using high-order spatial differencing on a nonstaggered grid. Mon. Wea. Rev.,116, 2069–2080.

  • ——, and ——, 1991: An efficient interpolation procedure for high-order three-dimensional semi-Lagrangian models. Mon. Wea. Rev.,119, 2492–2498.

  • ——, and ——, 1994: An efficient semi-Lagrangian scheme using third-order semi-implicit time integration and forward trajectories. Mon. Wea. Rev.,122, 745–756.

  • ——, and ——, 1996: Generalized Adams–Bashforth time integration schemes for a semi-Lagrangian model employing the second-derivative form of the horizontal momentum equations. Quart. J. Roy. Meteor. Soc.,122, 737–763.

  • Ritchie, H., 1991: Application of the semi-Lagrangian method to a multilevel spectral primitive-equations model. Quart. J. Roy. Meteor. Soc.,117, 91–106.

  • ——, C. Temperton, A. Simmons, M. Hortal, T. Davies, D. Dent, and M. Hamrud, 1995: Implementation of the semi-Lagrangian method in a high-resolution version of the ECMWF forecast model. Mon. Wea. Rev.,123, 489–514.

  • Robert, A., 1981: A stable numerical integration scheme for the primitive meteorological equations. Atmos.–Ocean,19, 35–46.

  • ——, 1982: A semi-Lagrangian and semi-implicit numerical integration scheme for the primitive meteorological equations. J. Meteor. Soc. Japan.,60, 319–325.

  • Staniforth, A. N., and C. Temperton, 1986: Semi-implicit semi-Lagrangian integration schemes for a barotropic finite-element regional model. Mon. Wea. Rev.,114, 2078–2090.

  • ——, and J. Côté, 1991: Semi-Lagrangian integration schemes for atmospheric models—A review. Mon. Wea. Rev.,119, 2206–2223.

  • Størmer, C., 1907: Sur les trajectoires des corpuscules électrisés (On the trajectories of charged corpuscles). Arch. Sci. Phys. Nat.,24, 5–8, 113–158, 221–247.

  • Temperton, C., and A. Staniforth, 1987: An efficient two-time-level semi-Lagrangian semi-implicit integration scheme. Quart. J. Roy. Meteor. Soc.,113, 1025–1039.

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 40 18 0
PDF Downloads 17 8 1

High-Order Generalized Lorenz N-Cycle Schemes for Semi-Lagrangian Models Employing Second Derivatives in Time

R. J. PurserGeneral Sciences Corporation, Laurel, Maryland, and NOAA/National Centers for Environmental Prediction, Washington, D.C.

Search for other papers by R. J. Purser in
Current site
Google Scholar
PubMed
Close
and
L. M. LeslieUniversity of New South Wales, Sydney, New South Wales, Australia

Search for other papers by L. M. Leslie in
Current site
Google Scholar
PubMed
Close
Restricted access

Abstract

Having recently demonstrated that significant enhancement of forecast accuracy in a semi-Lagrangian model results from the application of high-order time integration methods to the second-derivative form of the equations governing the trajectories, the authors here extend the range of available methods by introducing a class of what they call “generalized Lorenz” (GL) schemes. These explicit GL schemes, like Lorenz’s “N-cycle” methods, which inspired them, achieve a high formal accuracy in time for linear systems at an economy of storage that is the theoretical optimum. They are shown to possess robustly stable and consistent semi-implicit modifications that allow the deepest (fastest) gravity waves to be treated implicitly, so that integrations can proceed efficiently with time steps considerably longer than would be possible in an Eulerian framework.

Tests of the GL methods are conducted using an ensemble of 360 forecast cases over the Australian region at high spatial resolution, verifying at 48 h against a control forecast employing time steps sufficiently short to render time truncation errors negligible. Compared with the performance of the best alternative semi-Lagrangian treatment of equivalent storage economy (a quasi-second-order generalized Adams–Bashforth method), our new GL methods produce significant improvements both in formal accuracy and in actual forecast skill.

Corresponding author address: Dr. R. J. Purser, National Centers for Environmental Prediction, W/NP2 WWB Room 207, Washington, DC 20233.

Abstract

Having recently demonstrated that significant enhancement of forecast accuracy in a semi-Lagrangian model results from the application of high-order time integration methods to the second-derivative form of the equations governing the trajectories, the authors here extend the range of available methods by introducing a class of what they call “generalized Lorenz” (GL) schemes. These explicit GL schemes, like Lorenz’s “N-cycle” methods, which inspired them, achieve a high formal accuracy in time for linear systems at an economy of storage that is the theoretical optimum. They are shown to possess robustly stable and consistent semi-implicit modifications that allow the deepest (fastest) gravity waves to be treated implicitly, so that integrations can proceed efficiently with time steps considerably longer than would be possible in an Eulerian framework.

Tests of the GL methods are conducted using an ensemble of 360 forecast cases over the Australian region at high spatial resolution, verifying at 48 h against a control forecast employing time steps sufficiently short to render time truncation errors negligible. Compared with the performance of the best alternative semi-Lagrangian treatment of equivalent storage economy (a quasi-second-order generalized Adams–Bashforth method), our new GL methods produce significant improvements both in formal accuracy and in actual forecast skill.

Corresponding author address: Dr. R. J. Purser, National Centers for Environmental Prediction, W/NP2 WWB Room 207, Washington, DC 20233.

Save