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1. Arbitrarily high-order accurate simulations of compressible rotationally constrained convection using a transfinite mapping on cubed-sphere grids

   https://doi.org/10.1063/5.0158146  

   Chen, Kuangxu ; Liang, Chunlei ; Wan, Minping ( August 2023 , Physics of Fluids)

   We present two major improvements over the Compressible High-ORder Unstructured Spectral difference (CHORUS) code published in Wang et al., “A compressible high-order unstructured spectral difference code for stratified convection in rotating spherical shells,” J. Comput. Phys. 290, 90–111 (2015). The new code is named CHORUS++ in this paper. Subsequently, we perform a series of efficient simulations for rotationally constrained convection (RCC) in spherical shells. The first improvement lies in the integration of the high-order spectral difference method with a boundary-conforming transfinite mapping on cubed-sphere grids, thus ensuring exact geometric representations of spherical surfaces on arbitrary sparse grids. The second improvement is on the adoption of higher-order elements (sixth-order) in CHORUS++ vs third-order elements for the original CHORUS code. CHORUS++ enables high-fidelity RCC simulations using sixth-order elements on very coarse grids. To test the accuracy and efficiency of using elements of different orders, CHORUS++ is applied to a laminar solar benchmark, which is characterized by columnar banana-shaped convective cells. By fixing the total number of solution degrees of freedom, the computational cost per time step remains unchanged. Nevertheless, using higher-order elements in CHORUS++ resolves components of the radial energy flux much better than using third-order elements. To obtain converged predictions, using sixth-order elements is 8.7 times faster than using third-order elements. This significant speedup allows global-scale fully compressible RCC simulations to reach equilibration of the energy fluxes on a small cluster of just 40 cores. In contrast, CHORUS simulations were performed by Wang et al. on supercomputers using approximately 10 000 cores. Using sixth-order elements in CHORUS++, we further carry out global-scale solar convection simulations with decreased rotational velocities. Interconnected networks of downflow lanes emerge and surround broader and weaker regions of upflow fields. A strong inward kinetic energy flux compensated by an enhanced outward enthalpy flux appears. These observations are all consistent with those published in the literature. Furthermore, CHORUS++ can be extended to magnetohydrodynamic simulations with potential applications to the hydromagnetic dynamo processes in the interiors of stars and planets.

    

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2. A Spectral/hp-Based Stabilized Solver with Emphasis on the Euler Equations

   https://doi.org/10.3390/fluids9010018  

   Ranjan, Rakesh ; Catabriga, Lucia ; Araya, Guillermo ( January 2024 , Fluids)

   The solution of compressible flow equations is of interest with many aerospace engineering applications. Past literature has focused primarily on the solution of Computational Fluid Dynamics (CFD) problems with low-order finite element and finite volume methods. High-order methods are more the norm nowadays, in both a finite element and a finite volume setting. In this paper, inviscid compressible flow of an ideal gas is solved with high-order spectral/hp stabilized formulations using uniform high-order spectral element methods. The Euler equations are solved with high-order spectral element methods. Traditional definitions of stabilization parameters used in conjunction with traditional low-order bilinear Lagrange-based polynomials provide diffused results when applied to the high-order context. Thus, a revision of the definitions of the stabilization parameters was needed in a high-order spectral/hp framework. We introduce revised stabilization parameters, τsupg, with low-order finite element solutions. We also reexamine two standard definitions of the shock-capturing parameter, δ: the first is described with entropy variables, and the other is the YZβ parameter. We focus on applications with the above introduced stabilization parameters and analyze an array of problems in the high-speed flow regime. We demonstrate spectral convergence for the Kovasznay flow problem in both L1 and L2 norms. We numerically validate the revised definitions of the stabilization parameter with Sod’s shock and the oblique shock problems and compare the solutions with the exact solutions available in the literature. The high-order formulation is further extended to solve shock reflection and two-dimensional explosion problems. Following, we solve flow past a two-dimensional step at a Mach number of 3.0 and numerically validate the shock standoff distance with results obtained from NASA Overflow 2.2 code. Compressible flow computations with high-order spectral methods are found to perform satisfactorily for this supersonic inflow problem configuration. We extend the formulation to solve the implosion problem. Furthermore, we test the stabilization parameters on a complex flow configuration of AS-202 capsule analyzing the flight envelope. The proposed stabilization parameters have shown robustness, providing excellent results for both simple and complex geometries.

    

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3. Techniques, Tricks, and Algorithms for Efficient GPU-Based Processing of Higher Order Hyperbolic PDEs

   https://doi.org/10.1007/s42967-022-00235-9  

   Subramanian, Sethupathy ; Balsara, Dinshaw S. ; Bhoriya, Deepak ; Kumar, Harish ( March 2023 , Communications on Applied Mathematics and Computation)

   Chi-Wang Shu (Ed.)

   GPU computing is expected to play an integral part in all modern Exascale supercomputers. It is also expected that higher order Godunov schemes will make up about a significant fraction of the application mix on such supercomputers. It is, therefore, very important to prepare the community of users of higher order schemes for hyperbolic PDEs for this emerging opportunity. Not every algorithm that is used in the space-time update of the solution of hyperbolic PDEs will take well to GPUs. However, we identify a small core of algorithms that take exceptionally well to GPU computing. Based on an analysis of available options, we have been able to identify weighted essentially non-oscillatory (WENO) algorithms for spatial reconstruction along with arbitrary derivative (ADER) algorithms for time extension followed by a corrector step as the winning three-part algorithmic combination. Even when a winning subset of algorithms has been identified, it is not clear that they will port seamlessly to GPUs. The low data throughput between CPU and GPU, as well as the very small cache sizes on modern GPUs, implies that we have to think through all aspects of the task of porting an application to GPUs. For that reason, this paper identifies the techniques and tricks needed for making a successful port of this very useful class of higher order algorithms to GPUs. Application codes face a further challenge—the GPU results need to be practically indistinguishable from the CPU results—in order for the legacy knowledge bases embedded in these applications codes to be preserved during the port of GPUs. This requirement often makes a complete code rewrite impossible. For that reason, it is safest to use an approach based on OpenACC directives, so that most of the code remains intact (as long as it was originally well-written). This paper is intended to be a one-stop shop for anyone seeking to make an OpenACC-based port of a higher order Godunov scheme to GPUs. We focus on three broad and high-impact areas where higher order Godunov schemes are used. The first area is computational fluid dynamics (CFD). The second is computational magnetohydrodynamics (MHD) which has an involution constraint that has to be mimetically preserved. The third is computational electrodynamics (CED) which has involution constraints and also extremely stiff source terms. Together, these three diverse uses of higher order Godunov methodology, cover many of the most important applications areas. In all three cases, we show that the optimal use of algorithms, techniques, and tricks, along with the use of OpenACC, yields superlative speedups on GPUs. As a bonus, we find a most remarkable and desirable result: some higher order schemes, with their larger operations count per zone, show better speedup than lower order schemes on GPUs. In other words, the GPU is an optimal stratagem for overcoming the higher computational complexities of higher order schemes. Several avenues for future improvement have also been identified. A scalability study is presented for a real-world application using GPUs and comparable numbers of high-end multicore CPUs. It is found that GPUs offer a substantial performance benefit over comparable number of CPUs, especially when all the methods designed in this paper are used. 

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4. A Local Algorithm for Structure-Preserving Graph Cut

   https://doi.org/10.1145/3097983.3098015  

   Zhou, Dawei ; Zhang, Si ; Yildirim, Mehmet Yigit ; Alcorn, Scott ; Tong, Hanghang ; Davulcu, Hasan ; He, Jingrui ( August 2017 , KDD)

   Nowadays, large-scale graph data is being generated in a variety of real-world applications, from social networks to co-authorship networks, from protein-protein interaction networks to road traffic networks. Many existing works on graph mining focus on the vertices and edges, with the first-order Markov chain as the underlying model. They fail to explore the high-order network structures, which are of key importance in many high impact domains. For example, in bank customer personally identifiable information (PII) networks, the star structures often correspond to a set of synthetic identities; in financial transaction networks, the loop structures may indicate the existence of money laundering. In this paper, we focus on mining user-specified high-order network structures and aim to find a structure-rich subgraph which does not break many such structures by separating the subgraph from the rest. A key challenge associated with finding a structure-rich subgraph is the prohibitive computational cost. To address this problem, inspired by the family of local graph clustering algorithms for efficiently identifying a low-conductance cut without exploring the entire graph, we propose to generalize the key idea to model high-order network structures. In particular, we start with a generic definition of high-order conductance, and define the high-order diffusion core, which is based on a high-order random walk induced by user-specified high-order network structure. Then we propose a novel High-Order Structure-Preserving LOcal Cut (HOSPLOC) algorithm, which runs in polylogarithmic time with respect to the number of edges in the graph. It starts with a seed vertex and iteratively explores its neighborhood until a subgraph with a small high-order conductance is found. Furthermore, we analyze its performance in terms of both effectiveness and efficiency. The experimental results on both synthetic graphs and real graphs demonstrate the effectiveness and efficiency of our proposed HOSPLOC algorithm. 

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5. Quantitative analysis of hillshed geomorphology and critical zone function: Raising the hillshed to watershed status

   https://doi.org/10.1130/B35724.1  

   Brecheisen, Zachary S. ; Richter, Daniel D. ; Moon, Seulgi ; Halpin, Patrick N. ( December 2021 , GSA Bulletin)

   Landscapes are frequently delineated by nested watersheds and river networks ranked via stream orders. Landscapes have only recently been delineated by their interfluves and ridge networks, and ordered based on their ridge connectivity. There are, however, few studies that have quantitatively investigated the connections between interfluve networks and landscape morphology and environmental processes. Here, we ordered hillsheds using methods complementary to traditional watersheds, via a hierarchical ordering of interfluves, and we defined hillsheds to be landscape surfaces from which soil is shed by soil creep or any type of hillslope transport. With this approach, we demonstrated that hillsheds are most useful for analyses of landscape structure and processes. We ordered interfluve networks at the Calhoun Critical Zone Observatory (CZO), a North American Piedmont landscape, and demonstrated how interfluve networks and associated hillsheds are related to landscape geomorphology and processes of land management and land-use history, accelerated agricultural gully erosion, and bedrock weathering depth (i.e., regolith depth). Interfluve networks were ordered with an approach directly analogous to that first proposed for ordering streams and rivers by Robert Horton in the GSA Bulletin in 1945. At the Calhoun CZO, low-order hillsheds are numerous and dominate most of the observatory’s ∼190 km2 area. Low-order hillsheds are relatively narrow with small individual areas, they have relatively steep slopes with high curvature, and they are relatively low in elevation. In contrast, high-order hillsheds are few, large in individual area, and relatively level at high elevation. Cultivation was historically abandoned by farmers on severely eroding low-order hillsheds, and in fact agriculture continues today only on high-order hillsheds. Low-order hillsheds have an order of magnitude greater intensity of gullying across the Calhoun CZO landscape than high-order hillsheds. In addition, although modeled regolith depth appears to be similar across hillshed orders on average, both maximum modeled regolith depth and spatial depth variability decrease as hillshed order increases. Land management, geomorphology, pedology, and studies of land-use change can benefit from this new approach pairing landscape structure and analyses. 

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