Optimal Accuracy in Semi-Lagrangian Models

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  • 1 Department of Marine, Earth and Atmospheric Sciences, and Department of Mathematics, North Carolina State University, Raleigh North Carolina
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Abstract

A one-dimensional semi-implicit semi-Lagrangian (SISL) linear model and a nonlinear SISL global shallow-water model are employed to investigate the sensitivity of the solutions on (i) the order of interpolation applied at the departure points, (ii) trajectory uncentering adopted in recent studies to suppress computational gravitational noise, and (iii) the optimal truncation conditions.

The linear model results show that the truncation errors associated with the semi-Lagrangian (SL) part of the SISL scheme dominate the error characteristics when the estimates of the departure point values are based on linear or quadratic polynomial Lagrangian interpolation. In these two cases, the rate of error growth is large and may significantly exceed the acceptable levels in operational numerical weather prediction. Application of cubic interpolation drastically reduces the overall truncation errors, and the accuracy is well within the acceptable range. In contrast with the schemes based on lower-order interpolation (linear or quadratic), the authors show that the truncation errors in the cubic interpolation case are almost entirely due to the semi-implicit (SI) part of the SISL scheme. It is apparent from the results that application of higher-order interpolation would not significantly improve the overall accuracy for fixed resolution in space and time. In the case of planetary-scale waves, uncentering tends to impose a more stringent upper limit on the choice of the size of time step compared to the fully centered scheme. Unlike the Eulerian numerical formulation case. this constraint arises from accuracy considerations rather than stability limitations. Therefore, although trajectory uncentering has recently been shown to be effective in suppressing undesirable gravitational computational noise, it must he applied cautiously as it may also degrade the low-frequency component of the flow that one wishes to preserve and predict. The synoptic-scale waves are relatively less sensitive to the damping associated with uncentering. For large-scale waves, use of coarser spatial resolution may even result in improved accuracy, provided that the size of the time step is chosen appropriately.

Turning to the shallow-water model formulation, two kinds of simulations are performed to investigate the accuracy of the semi-Lagrangian numerical scheme. The method of prescribed solutions is adopted by introducing a forcing term in the mass continuity equation such that Rossby-Haurwitz functions are exact analytic solutions to the modified form of the shallow-water equations. We adopt the Arakawa C grid with a uniform grid size in the zonal and meridional directions of approximately 1.4°. The initial conditions are composed of planetary-scale Rossby-Haurwitz solutions and dominated by wavenumber 4 (R = 4). The first category of experiments consists of ten 5-day simulations. The Rossby-Haurwitz “exact” solutions for the modified shallow-water equations are adopted as the reference fields for assessing the model performance. Therefore, any departure of the numerical solution from the analytic solution represents a measure of the model error. As the size of the time step is increased, first the error decreases until it attains a minimum at approximately Δt = 30 min and then it begins to increase monotonically with further increase in the size of the time step. This development is also consistent with the results based on the analysis of the linear model. The second kind of experiment consists of another set of ten 5-day simulations based on the traditional identical-twin model design approach in which the control or “truth” run is based on a small time step of 10 min. In the rest of the runs under this category, the spatial resolution is again held fixed while the size of time step is progressively increased in each subsequent simulation from Δt = 10 min up to Δt = 120 min. The results show a monotonic relationship between the size of the time step and the global average rms error at day 5, with the larger time steps exhibiting greater error. We observe a regime of minimum error growth with increasing size of the time step in the intermediate range between 0.5 and 1.0 h. Based on the comparison between the results from the two categories of simulations, we infer that the results based on the identical twin experiment design could portray a misleading representation of the true truncation errors. We further note that the range of the size of time steps corresponding to the minimum numerical truncation errors in the first category of simulations coincides with the range of time steps, which exhibit minimum error truncation growth associated with the second category of experiments. This may eventually lead to the development of a practical procedure for identifying the optimal truncation conditions when the exact solutions to the governing system of equations are not known.

Abstract

A one-dimensional semi-implicit semi-Lagrangian (SISL) linear model and a nonlinear SISL global shallow-water model are employed to investigate the sensitivity of the solutions on (i) the order of interpolation applied at the departure points, (ii) trajectory uncentering adopted in recent studies to suppress computational gravitational noise, and (iii) the optimal truncation conditions.

The linear model results show that the truncation errors associated with the semi-Lagrangian (SL) part of the SISL scheme dominate the error characteristics when the estimates of the departure point values are based on linear or quadratic polynomial Lagrangian interpolation. In these two cases, the rate of error growth is large and may significantly exceed the acceptable levels in operational numerical weather prediction. Application of cubic interpolation drastically reduces the overall truncation errors, and the accuracy is well within the acceptable range. In contrast with the schemes based on lower-order interpolation (linear or quadratic), the authors show that the truncation errors in the cubic interpolation case are almost entirely due to the semi-implicit (SI) part of the SISL scheme. It is apparent from the results that application of higher-order interpolation would not significantly improve the overall accuracy for fixed resolution in space and time. In the case of planetary-scale waves, uncentering tends to impose a more stringent upper limit on the choice of the size of time step compared to the fully centered scheme. Unlike the Eulerian numerical formulation case. this constraint arises from accuracy considerations rather than stability limitations. Therefore, although trajectory uncentering has recently been shown to be effective in suppressing undesirable gravitational computational noise, it must he applied cautiously as it may also degrade the low-frequency component of the flow that one wishes to preserve and predict. The synoptic-scale waves are relatively less sensitive to the damping associated with uncentering. For large-scale waves, use of coarser spatial resolution may even result in improved accuracy, provided that the size of the time step is chosen appropriately.

Turning to the shallow-water model formulation, two kinds of simulations are performed to investigate the accuracy of the semi-Lagrangian numerical scheme. The method of prescribed solutions is adopted by introducing a forcing term in the mass continuity equation such that Rossby-Haurwitz functions are exact analytic solutions to the modified form of the shallow-water equations. We adopt the Arakawa C grid with a uniform grid size in the zonal and meridional directions of approximately 1.4°. The initial conditions are composed of planetary-scale Rossby-Haurwitz solutions and dominated by wavenumber 4 (R = 4). The first category of experiments consists of ten 5-day simulations. The Rossby-Haurwitz “exact” solutions for the modified shallow-water equations are adopted as the reference fields for assessing the model performance. Therefore, any departure of the numerical solution from the analytic solution represents a measure of the model error. As the size of the time step is increased, first the error decreases until it attains a minimum at approximately Δt = 30 min and then it begins to increase monotonically with further increase in the size of the time step. This development is also consistent with the results based on the analysis of the linear model. The second kind of experiment consists of another set of ten 5-day simulations based on the traditional identical-twin model design approach in which the control or “truth” run is based on a small time step of 10 min. In the rest of the runs under this category, the spatial resolution is again held fixed while the size of time step is progressively increased in each subsequent simulation from Δt = 10 min up to Δt = 120 min. The results show a monotonic relationship between the size of the time step and the global average rms error at day 5, with the larger time steps exhibiting greater error. We observe a regime of minimum error growth with increasing size of the time step in the intermediate range between 0.5 and 1.0 h. Based on the comparison between the results from the two categories of simulations, we infer that the results based on the identical twin experiment design could portray a misleading representation of the true truncation errors. We further note that the range of the size of time steps corresponding to the minimum numerical truncation errors in the first category of simulations coincides with the range of time steps, which exhibit minimum error truncation growth associated with the second category of experiments. This may eventually lead to the development of a practical procedure for identifying the optimal truncation conditions when the exact solutions to the governing system of equations are not known.

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