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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. Efficient WENO-Based Prolongation Strategies for Divergence-Preserving Vector Fields

   https://doi.org/10.1007/s42967-021-00182-x  

   Balsara, Dinshaw S.; Samantaray, Saurav; Subramanian, Sethupathy (May 2022, Communications on Applied Mathematics and Computation)

   Abstract Adaptive mesh refinement (AMR) is the art of solving PDEs on a mesh hierarchy with increasing mesh refinement at each level of the hierarchy. Accurate treatment on AMR hierarchies requires accurate prolongation of the solution from a coarse mesh to a newly defined finer mesh. For scalar variables, suitably high-order finite volume WENO methods can carry out such a prolongation. However, classes of PDEs, such as computational electrodynamics (CED) and magnetohydrodynamics (MHD), require that vector fields preserve a divergence constraint. The primal variables in such schemes consist of normal components of the vector field that are collocated at the faces of the mesh. As a result, the reconstruction and prolongation strategies for divergence constraint-preserving vector fields are necessarily more intricate. In this paper we present a fourth-order divergence constraint-preserving prolongation strategy that is analytically exact. Extension to higher orders using analytically exact methods is very challenging. To overcome that challenge, a novel WENO-like reconstruction strategy is invented that matches the moments of the vector field in the faces, where the vector field components are collocated. This approach is almost divergence constraint-preserving, therefore, we call it WENO-ADP. To make it exactly divergence constraint-preserving, a touch-up procedure is developed that is based on a constrained least squares (CLSQ) method for restoring the divergence constraint up to machine accuracy. With the touch-up, it is called WENO-ADPT. It is shown that refinement ratios of two and higher can be accommodated. An item of broader interest in this work is that we have also been able to invent very efficient finite volume WENO methods, where the coefficients are very easily obtained and the multidimensional smoothness indicators can be expressed as perfect squares. We demonstrate that the divergence constraint-preserving strategy works at several high orders for divergence-free vector fields as well as vector fields, where the divergence of the vector field has to match a charge density and its higher moments. We also show that our methods overcome the late time instability that has been known to plague adaptive computations in CED. 

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3. Improving Inference for Neural Image Compression

   Yang, Yibo; Bamler, Robert; Mandt, Stephan (October 2020, Advances in neural information processing systems)

   null (Ed.)

   We consider the problem of lossy image compression with deep latent variable models. State-of-the-art methods [Ballé et al., 2018, Minnen et al., 2018, Lee et al., 2019] build on hierarchical variational autoencoders (VAEs) and learn inference networks to predict a compressible latent representation of each data point. Drawing on the variational inference perspective on compression [Alemi et al., 2018], we identify three approximation gaps which limit performance in the conventional approach: an amortization gap, a discretization gap, and a marginalization gap. We propose remedies for each of these three limitations based on ideas related to iterative inference, stochastic annealing for discrete optimization, and bits-back coding, resulting in the first application of bits-back coding to lossy compression. In our experiments, which include extensive baseline comparisons and ablation studies, we achieve new state-of-the-art performance on lossy image compression using an established VAE architecture, by changing only the inference method. 

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4. Steady translating hollow vortex pair in weakly compressible flow

   https://doi.org/10.1016/j.physd.2023.133943  

   Krishnamurthy, Vikas S.; Llewellyn Smith, Stefan G. (January 2024, Physica D: Nonlinear Phenomena)

   The weakly compressible translating hollow vortex pair is examined using the Imai–Lamla formula and a direct conformal mapping approach applied to a Rayleigh–Jansen expansion in the Mach number, 𝑀, taken to be small. The incompressible limit has been studied by Crowdy et al. (Eur. J. Mech. B Fluids 37 (2013), 180–186) who found explicit formulas for the solutions using conformal mapping. We improve upon their results by finding an explicit formula for the conformal map which had been left as an integral in Crowdy et al. (2013). The weakly compressible problem requires the solution of two boundary value problems in an annulus. This results in a linear parameter problem for the perturbed propagation velocity and speed along the vortex boundary. We find that two additional constraints are required to solve for the perturbed parameters. We require that the perturbation in vortex area and the perturbation in centroid separation to vanish. Three possible centroid definitions are given and the results worked out for each case. It is found that the correction to the propagation velocity is always negative, so that the vortex always slows down at first order due to compressible effects. Numerical results are consistent in the limit of small vortex size with the previous result by Leppington (J. Fluid Mech. 559 (2006), 45–55) that the propagation parameter is unchanged to 𝑂(𝑀2). 

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5. Comparison of adaptive mesh refinement techniques for numerical weather prediction

   Abdi, Daniel S; Almgren, Ann; Giraldo, Francis X; Jankov, Isidora (April 2024, arXiv but under review in Monthly Weather Review)

   This paper examines the application of adaptive mesh refinement (AMR) in the field of numerical weather prediction (NWP). We implement and assess two distinct AMR approaches and evaluate their performance through standard NWP benchmarks. In both cases, we solve the fully compressible Euler equations, fundamental to many non-hydrostatic weather models. The first approach utilizes oct-tree cell-based mesh refinement coupled with a high-order discontinuous Galerkin method for spatial discretization. In the second approach, we employ level-based AMR with the finite difference method. Our study provides insights into the accuracy and benefits of employing these AMR methodologies for the multi-scale problem of NWP. Additionally, we explore essential properties including their impact on mass and energy conservation. Moreover, we present and evaluate an AMR solution transfer strategy for the tree-based AMR approach that is simple to implement, memory-efficient, and ensures conservation for both flow in the box and sphere. Furthermore, we discuss scalability, performance portability, and the practical utility of the AMR methodology within an NWP framework -- crucial considerations in selecting an AMR approach. The current de facto standard for mesh refinement in NWP employs a relatively simplistic approach of static nested grids, either within a general circulation model or a separately operated regional model with loose one-way synchronization. It is our hope that this study will stimulate further interest in the adoption of AMR frameworks like AMReX in NWP. These frameworks offer a triple advantage: a robust dynamic AMR for tracking localized and consequential features such as tropical cyclones, extreme scalability, and performance portability. 

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