the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 18 Aug 2023
Sweep Interpolation: A Fourth-Order Accurate Cost Effective Scheme in the Global Environmental Multiscale Model
Mohammad Mortezazadeh
Jean-Francois Cossette
Ashu Dastoor
Jean de Grandpré
Irena Ivanova
Abdessamad Qaddouri
Abstract. The interpolation process is the most computationally expensive step of the semi-Lagrangian (SL) approach for solving advection which is commonly used in numerical weather prediction (NWP) models. It has a significant impact on the accuracy of the solution and can potentially be the most expensive part of model integration. The sweep algorithm, which was first described by Mortezazadeh and Wang (2017), performs SL interpolation with the same computational cost as a third order polynomial scheme but with the accuracy of a fourth order interpolation scheme. This improvement is achieved by using two 3rd-order backward and forward polynomial interpolation schemes in two consecutive time steps. In this paper, we present a new application of the sweep algorithm within the context of global forecasts produced with Environment Climate Change Canada’s Global Environmental Multiscale (GEM) model. Results show that the sweep interpolation scheme is computationally more efficient compared to a conventional fourth order polynomial scheme, especially evident for increasing number of several advected several passive tracers. An additional advantage of this new approach is that its implementation in a chemical and weather forecast models requires minimum modifications of the interpolation weighting coefficients. An analysis of the computational performance for a set of theoretical benchmarks as well as a global ozone forecast experiment show that up to 15 % reduction in total wall clock time is achieved. Forecasting experiments using the global version of the GEM model and the new interpolation show that the sweep interpolation can perform very well in predicting ozone distribution, especially in the tropopause region where transport processes play a significant role.
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Notice on discussion status
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
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Preprint
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The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
- Preprint
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- Final revised paper
Journal article(s) based on this preprint
Mohammad Mortezazadeh et al.
Interactive discussion
Status: closed
-
CC1: 'Comment on egusphere-2023-1508', Jixiang Li, 21 Sep 2023
Thank you for sharing your research. This paper presents a fourth-order accurate and cost-effective scheme called sweep interpolation, which uses fewer neighboring cells than the cubic interpolation. It significantly reduces computational time while maintaining very close accuracy to the typical fourth-order interpolation. However, there are still some issues that need to be addressed before it can be accepted for publication in GMD.
(1) Different interpolation schemes should have different contents. Please compare the differences in contour maps between CUBIC and SWEEP interpolations in Figure 3, Figure 4, and Figure 5.
(2) In the atmospheric methane-like tracer test case, the differences between cubic and sweep interpolations are apparent (Figure 5b), and the reasons for these differences should be analyzed.
(3) Serial numbers are not marked in Figures 4 and 5.
(4) Place all the pictures on one page in Figures 4 and 6.
Citation: https://doi.org/10.5194/egusphere-2023-1508-CC1 -
AC1: 'Reply on CC1', Mohammad Mortezazadeh, 09 Nov 2023
Dear Dr Li,
Thank you for your keen interest in our work and for your valuable and thoughtful feedback.
Many of the concerns you've highlighted have also been addressed in response to other reviewers. The attached document specifies the responses to your comments.
Best regards,
Mohammad Mortezazadeh and co-authors.
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AC1: 'Reply on CC1', Mohammad Mortezazadeh, 09 Nov 2023
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RC1: 'Comment on egusphere-2023-1508', Li Dong, 26 Sep 2023
General comments:
This manuscript describes the application of the sweep interpolation with fourth-order accuracy in the GEM. As we all know, the interpolation algorithm for the velocity and tracer densities is vital for the semi-Lagrangian method. The authors proposed an elaborate idea of combining two interpolation stencils to cancel the leading errors. The sweep algorithm is efficient as the third-order one but with higher accuracy as the fourth-order one, and it is easy to implement. The numerical experiment results illustrate the efficacy of the sweep interpolation algorithm. I recommend the publication of this manuscript subject to a minor revision.
Specific comments:
It would be better that some details can be further explained:
- In 2D, there are four possible stencil combinations as shown in Fig. 1 of Mortezazadeh and Wang (2017). Is the selection of forward and backward interpolation stencils related to the parcel characteristic line? Or if the two stencils change according to the backward trajectory?
- The description of the tests is too brief, such as sec. 3.1. Please add more information, such as what the spatial resolution is?
- Why the total mass of sweep scheme is decreasing, while cubic scheme is increasing in Fig. 5.
Technical corrections:
L77: The variable staggering in the vertical direction is the Charney-Phillips grid, so it should not be the Arackawa-C grid in the vertical direction.
L83: “sub grids” to “subgrids”
L116: “ ,on 1D” to “, along the 1D direction”?
Citation: https://doi.org/10.5194/egusphere-2023-1508-RC1 -
AC2: 'Reply on RC1', Mohammad Mortezazadeh, 09 Nov 2023
Dear Dr Dong,
We would like to extend our sincere appreciation to the reviewer for the positive and constructive feedback on our manuscript.
Here in the following, we attached our responses to your comments.
best,
Mohammad Mortezazadeh and co-authors.
-
RC2: 'Comment on egusphere-2023-1508', Xiaqiong Zhou, 09 Oct 2023
This work extends the research conducted by Mortezazadeh and Wang in 2017. In this study, the sweep method is further validated through a series of idealized tests, including 2D vortex simulations and Hadley-like meridional circulation, as well as an Atmospheric methane-like tracer test and global model forecasts. This manuscript demonstrates that the sweep method can significantly reduce computational costs by approximately 15% without compromising accuracy. This improvement is achieved by implementing two 3rd-order backward and forward polynomial interpolation schemes over two consecutive time steps, as opposed to using a 4th-order interpolation method. The results presented in this manuscript are intriguing and robust, supporting its acceptance for publication with only minor revisions.
- In Figure 3(b) and 4(b), the black dotted line represents the mass error after every two time steps. To enhance clarity, consider changing the label from "Error after two time steps" to "Error every two time steps."
- It is essential to delve into the motivation behind and the conclusions drawn from the 2D vortex tests. Further discussions are warranted to provide a comprehensive understanding of their significance.
- Figure 3a illustrates that the error distribution of the 2D vortex simulations is less noisy outside the vortex region when using the sweep method. Is this observed reduction in noise attributed to the method's capability to minimize dispersion and dissipation errors? Additional clarification on this matter would be beneficial.
- Section 3.3, pertaining to the Atmospheric methane-like tracer experiment, lacks a clear description of the experiment's design. While information on resolution and time steps is provided, it would be beneficial to include more detailed descriptions.
Citation: https://doi.org/10.5194/egusphere-2023-1508-RC2 -
AC3: 'Reply on RC2', Mohammad Mortezazadeh, 09 Nov 2023
Dear Dr. Zhou,
Thank you for your thoughtful consideration and positive response.
Your comments and suggestions will undoubtedly contribute to improving the manuscript.
here we attached our responses to your comments.
Best,
Mohammad Mortezazadeh and co-authors.
Interactive discussion
Status: closed
-
CC1: 'Comment on egusphere-2023-1508', Jixiang Li, 21 Sep 2023
Thank you for sharing your research. This paper presents a fourth-order accurate and cost-effective scheme called sweep interpolation, which uses fewer neighboring cells than the cubic interpolation. It significantly reduces computational time while maintaining very close accuracy to the typical fourth-order interpolation. However, there are still some issues that need to be addressed before it can be accepted for publication in GMD.
(1) Different interpolation schemes should have different contents. Please compare the differences in contour maps between CUBIC and SWEEP interpolations in Figure 3, Figure 4, and Figure 5.
(2) In the atmospheric methane-like tracer test case, the differences between cubic and sweep interpolations are apparent (Figure 5b), and the reasons for these differences should be analyzed.
(3) Serial numbers are not marked in Figures 4 and 5.
(4) Place all the pictures on one page in Figures 4 and 6.
Citation: https://doi.org/10.5194/egusphere-2023-1508-CC1 -
AC1: 'Reply on CC1', Mohammad Mortezazadeh, 09 Nov 2023
Dear Dr Li,
Thank you for your keen interest in our work and for your valuable and thoughtful feedback.
Many of the concerns you've highlighted have also been addressed in response to other reviewers. The attached document specifies the responses to your comments.
Best regards,
Mohammad Mortezazadeh and co-authors.
-
AC1: 'Reply on CC1', Mohammad Mortezazadeh, 09 Nov 2023
-
RC1: 'Comment on egusphere-2023-1508', Li Dong, 26 Sep 2023
General comments:
This manuscript describes the application of the sweep interpolation with fourth-order accuracy in the GEM. As we all know, the interpolation algorithm for the velocity and tracer densities is vital for the semi-Lagrangian method. The authors proposed an elaborate idea of combining two interpolation stencils to cancel the leading errors. The sweep algorithm is efficient as the third-order one but with higher accuracy as the fourth-order one, and it is easy to implement. The numerical experiment results illustrate the efficacy of the sweep interpolation algorithm. I recommend the publication of this manuscript subject to a minor revision.
Specific comments:
It would be better that some details can be further explained:
- In 2D, there are four possible stencil combinations as shown in Fig. 1 of Mortezazadeh and Wang (2017). Is the selection of forward and backward interpolation stencils related to the parcel characteristic line? Or if the two stencils change according to the backward trajectory?
- The description of the tests is too brief, such as sec. 3.1. Please add more information, such as what the spatial resolution is?
- Why the total mass of sweep scheme is decreasing, while cubic scheme is increasing in Fig. 5.
Technical corrections:
L77: The variable staggering in the vertical direction is the Charney-Phillips grid, so it should not be the Arackawa-C grid in the vertical direction.
L83: “sub grids” to “subgrids”
L116: “ ,on 1D” to “, along the 1D direction”?
Citation: https://doi.org/10.5194/egusphere-2023-1508-RC1 -
AC2: 'Reply on RC1', Mohammad Mortezazadeh, 09 Nov 2023
Dear Dr Dong,
We would like to extend our sincere appreciation to the reviewer for the positive and constructive feedback on our manuscript.
Here in the following, we attached our responses to your comments.
best,
Mohammad Mortezazadeh and co-authors.
-
RC2: 'Comment on egusphere-2023-1508', Xiaqiong Zhou, 09 Oct 2023
This work extends the research conducted by Mortezazadeh and Wang in 2017. In this study, the sweep method is further validated through a series of idealized tests, including 2D vortex simulations and Hadley-like meridional circulation, as well as an Atmospheric methane-like tracer test and global model forecasts. This manuscript demonstrates that the sweep method can significantly reduce computational costs by approximately 15% without compromising accuracy. This improvement is achieved by implementing two 3rd-order backward and forward polynomial interpolation schemes over two consecutive time steps, as opposed to using a 4th-order interpolation method. The results presented in this manuscript are intriguing and robust, supporting its acceptance for publication with only minor revisions.
- In Figure 3(b) and 4(b), the black dotted line represents the mass error after every two time steps. To enhance clarity, consider changing the label from "Error after two time steps" to "Error every two time steps."
- It is essential to delve into the motivation behind and the conclusions drawn from the 2D vortex tests. Further discussions are warranted to provide a comprehensive understanding of their significance.
- Figure 3a illustrates that the error distribution of the 2D vortex simulations is less noisy outside the vortex region when using the sweep method. Is this observed reduction in noise attributed to the method's capability to minimize dispersion and dissipation errors? Additional clarification on this matter would be beneficial.
- Section 3.3, pertaining to the Atmospheric methane-like tracer experiment, lacks a clear description of the experiment's design. While information on resolution and time steps is provided, it would be beneficial to include more detailed descriptions.
Citation: https://doi.org/10.5194/egusphere-2023-1508-RC2 -
AC3: 'Reply on RC2', Mohammad Mortezazadeh, 09 Nov 2023
Dear Dr. Zhou,
Thank you for your thoughtful consideration and positive response.
Your comments and suggestions will undoubtedly contribute to improving the manuscript.
here we attached our responses to your comments.
Best,
Mohammad Mortezazadeh and co-authors.
Peer review completion
Journal article(s) based on this preprint
Mohammad Mortezazadeh et al.
Mohammad Mortezazadeh et al.
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The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
- Preprint
(1981 KB) - Metadata XML