Assessing the Impact of High-Order Curved Elements on Simulations of Atmospheric Flows over Complex Orography using a Discontinuous Galerkin Scheme

Core Concepts

The use of high-order curved elements significantly improves the accuracy of numerical simulations of atmospheric flows over complex orography compared to linear mapping, especially for non-smooth orographic profiles.

Abstract

The authors present a quantitative assessment of the impact of high-order mappings on the simulation of flows over complex orography using a Discontinuous Galerkin (DG) method. Key highlights: Curved boundaries are increasingly employed in combination with high-order methods like the Finite Element Method (FEM) and Spectral Element Method (SEM) to maintain high-order accuracy. The authors consider a DG method implemented in the deal.II library, which natively supports high-order mappings. Numerical experiments on classical benchmarks over idealized orographic profiles demonstrate the positive impact of curved boundaries on the accuracy of the results. For smooth orographic profiles, high-order mappings lead to more accurate results than linear mapping. High-order mappings also provide similar results to higher resolution simulations with linear mapping. For non-smooth orographic profiles, high-order mappings outperform linear mapping, even when the non-smooth features are filtered. The use of high-order mappings captures small-scale topographic features that are inaccessible to linear mapping at the same resolution. The authors also assess the impact of high-order mappings on a realistic complex orography, showing improved accuracy compared to linear mapping. Overall, the results highlight the importance of using high-order curved elements to accurately represent complex orography in atmospheric flow simulations.

Stats

The momentum flux is a key diagnostic quantity used to analyze the orographic response. The authors compute the momentum flux using two different formulas: m(z) = -π/4 ρs us N hm^2 (linear theory) m(z) = ∫ (ρ + ρ') (u + u') w' dx (full vertical flux)

Quotes

"High-order numerical methods are employed to an increasing extent for many relevant physical problems, in particular for applications in Computational Fluid Dynamics (CFD), see e.g. [6, 16, 20] among many others." "While designing numerical schemes in combination with curved boundaries posed limitations for early numerical methods, a number of approaches to deal naturally with curved geometries have been proposed for high-order methods over the last fifty years."

Key Insights Distilled From

Impact of curved elements for flows over orography with a Discontinuous Galerkin scheme

by Giuseppe Orl... at arxiv.org 04-16-2024

https://arxiv.org/pdf/2404.09319.pdf

Deeper Inquiries

How can the insights from this study on the importance of high-order curved elements be extended to other applications beyond atmospheric flows, such as in engineering or biomedical simulations

The insights from the study on the importance of high-order curved elements in atmospheric flows can be extended to various other applications beyond meteorology. In engineering simulations, such as fluid dynamics or structural analysis, the accurate representation of complex geometries is crucial for obtaining reliable results. By using high-order curved elements, engineers can capture intricate details of the geometry, leading to more precise simulations and better understanding of the physical phenomena involved. For example, in aerodynamics, the use of high-order curved elements can improve the accuracy of airflow simulations around aircraft or vehicles, leading to optimized designs and improved performance. In biomedical simulations, such as modeling blood flow in arteries or simulating the behavior of biological tissues, the use of high-order curved elements can provide a more realistic representation of the complex geometries involved. This can lead to better insights into the biomechanical behavior of the systems under study, aiding in the development of medical devices or treatment strategies. Additionally, in computational biology, high-order curved elements can help in modeling intricate biological structures with greater accuracy, leading to more reliable predictions and insights into biological processes. Overall, the insights from the study on high-order curved elements in atmospheric flows can be applied to a wide range of applications in engineering and biomedical simulations, where accurate representation of complex geometries is essential for obtaining meaningful results.

What are the computational trade-offs and challenges in implementing high-order curved elements compared to linear mapping, and how can these be addressed to make the approach more practical for large-scale simulations

Implementing high-order curved elements in simulations can offer significant advantages in terms of accuracy and precision, but it also comes with computational trade-offs and challenges compared to linear mapping. One of the main challenges is the increased computational cost associated with high-order elements, as they require more computational resources and memory to handle the additional complexity. This can lead to longer simulation times and higher hardware requirements, making the approach less practical for large-scale simulations. To address these challenges and make the approach more practical, several strategies can be employed. One approach is to optimize the implementation of high-order curved elements by leveraging parallel computing techniques and efficient algorithms to reduce computational overhead. Additionally, adaptive mesh refinement techniques can be used to selectively increase the resolution in areas where high-order elements are necessary, while maintaining lower resolutions in less critical regions to save computational resources. Furthermore, advancements in hardware technology, such as the use of high-performance computing systems or specialized hardware accelerators, can help mitigate the computational challenges associated with high-order curved elements. By leveraging these technologies, simulations with high-order curved elements can be made more feasible for large-scale applications, enabling researchers and engineers to benefit from the increased accuracy and precision offered by such approaches.

Given the sensitivity of the results to the choice of filtering parameters for non-smooth orography, how can the interplay between resolved and parameterized orographic effects be better understood and optimized for improved weather and climate modeling

The sensitivity of the results to the choice of filtering parameters for non-smooth orography highlights the importance of understanding and optimizing the interplay between resolved and parameterized orographic effects in weather and climate modeling. To better address this issue, researchers can focus on the following strategies: Sensitivity Analysis: Conducting sensitivity analyses to understand how different filtering parameters impact the simulation results can provide valuable insights into the interplay between resolved and parameterized effects. By systematically varying the filtering parameters and analyzing the resulting changes in the simulations, researchers can identify optimal settings that balance accuracy and computational efficiency. Model Calibration: Calibrating the filtering parameters based on observational data or benchmark simulations can help improve the performance of the model. By comparing the model outputs with real-world observations or validated simulations, researchers can adjust the filtering parameters to better match the expected behavior of the system. Hybrid Approaches: Exploring hybrid approaches that combine resolved and parameterized orographic effects in a seamless manner can help optimize the overall simulation performance. By integrating advanced filtering techniques with adaptive mesh refinement strategies or data assimilation methods, researchers can enhance the accuracy of the simulations while maintaining computational efficiency. Collaborative Research: Collaborating with experts in atmospheric science, computational modeling, and data analysis can provide interdisciplinary perspectives on optimizing the interplay between resolved and parameterized orographic effects. By bringing together diverse expertise, researchers can develop innovative solutions to address the challenges associated with non-smooth orography in weather and climate modeling. By implementing these strategies and fostering collaboration across disciplines, researchers can better understand and optimize the interplay between resolved and parameterized orographic effects, leading to improved weather and climate modeling capabilities.