arXiv subjects

Anna Schwarz

Publications and source records attributed to Anna Schwarz.

16 recordsLinked to original sources

arXiv2026

h-Adaptive FV Subcell Shock-Capturing for DGSEM on Heterogeneous Curvilinear Meshes

High-order methods offer superior dispersion and dissipation properties compared to low-order schemes but require robust stabilization for discontinuities. To ensure stability, local artificial viscosity is common, but often degrades sub-element resolution. Conversely, subcell resolution preserving limiting strategies such as the finite volume subcell method are typically restricted to uniform topologies, such as purely hexahedral or simplex meshes, or linear elements. This leaves a significant gap in treating the hybrid-element topologies necessary for complex engineering geometries. To bridge this gap, we introduce a robust shock-capturing approach for the discontinuous Galerkin spectral element method on mixed curvilinear meshes containing hexahedral, prismatic, tetrahedral, and pyramid elements. Non-hexahedral elements are handled via collapsed coordinate transformations. The proposed method utilizes an $h$-adaptive finite volume subcell scheme with an arbitrary subcell resolution up to $2\mathcal{N}+1$. Special care is taken to ensure conforming subcell distributions across different element types. Crucially, discrete conservation is proven theoretically and verified numerically, alongside validations for high-order convergence and robust shock capturing. Finally, the method's applicability to complex configurations is demonstrated through a simulation of the flow around a NACA 0012 airfoil.

math.NA↗

arXiv2026

APU-Accelerated Large Eddy Simulation with the Discontinuous Galerkin Solver GALÆXI

The exascale computing era, driven by heterogeneous GPU architectures, requires a fundamental redesign of traditional CFD solvers to fully leverage those heterogeneous systems. The discontinuous Galerkin spectral element method (DGSEM) provides an ideal foundation for this transition due to its high-order accuracy and local computational stencil. This work presents recent advances in the development and application of the architecture-agnostic DGSEM framework GALÆXI by linking hardware optimization, software implementation, and physical validation. The performance of GALÆXI on the AMD MI300A Accelerated Processing Units (APUs) featured on the Hunter supercomputer is analyzed. Specifically, evaluations of the strong and weak scaling performance and the impact of the compute partitioning modes available on the AMD MI300As are performed. Second, the strategy used to integrate the algorithms necessary for wall-modeled large eddy simulations into the GPU-accelerated framework is outlined. Validation of those algorithms is presented in the form of a plane turbulent channel testcase. Finally, the solver is applied to a demanding flow problem in the form of a wall-resolved large eddy simulation of a transonic compressor cascade. The results from this investigation demonstrate the capabilities of GALÆXI to accurately capture complex shock-wave/turbulent boundary-layer interactions.

physics.flu-dyn↗

arXiv2026

High-Order Simulation of Particle-Laden Flows in Moving Domains Using Coupled ALE and Sliding Mesh Approaches

In practical applications, compressible particle-laden flows in moving geometries involve complex, non-linear, and multi-scale inter-actions with turbulent structures. Resolving these dynamics numerically requires careful algorithmic treatment to accurately predict particle trajectories. This work presents a high-fidelity Euler-Lagrange framework that couples a high-order discontinuous Galerkin spectral element method for the continuous phase with a Lagrangian point-particle tracking scheme. To manage moving and deforming domains, the framework integrates two distinct mesh movement strategies: the arbitrary Lagrangian-Eulerian method for general mesh deformations such as time-resolved particle-induced surface deformations and its special interface case, the sliding mesh approach, uniquely suited for rigid rotational or translational movements. A primary focus is placed on tightly coupling the arbitrary Lagrangian-Eulerian formulation into the temporal evolution step by utilizing radial basis function morphing to capture the non-linear feedback loop between evolving surface topologies and the continuous phase. Concurrently, the framework ensures time- and high-order accurate coupling of the mesh movement algorithms with the dispersed phase. In particular, the proposed algorithm resolves the sliding mesh tracking problem by enforcing strict spatial and temporal accuracy as Lagrangian particles cross non-conforming grid interfaces between adjacent moving zones. The algorithms are rigorously validated against multiple benchmarks and subsequently applied to two challenging compressor rotor applications: the first focusing on solid-particle erosion, and the second featuring an upstream cylindrical wake generator.

physics.comp-ph↗

arXiv2026

An Architecture-Agnostic High-Order Discontinuous Galerkin Framework for Compressible Flows

With the recent proliferation of heterogeneous, GPU-accelerated supercomputers, high-order computational fluid dynamics (CFD) simulations of complex, turbulent flows are more accessible than ever. To leverage the computing power of these machines, CFD software must adapt. However, complicating the situation is the emerging need to support hardware from multiple GPU vendors. Addressing this need is the GPU-accelerated, discontinuous Galerkin spectral element method (DGSEM) framework GALÆXI, a high-order, open source, architecture-agnostic toolchain for the study of complex, compressible, turbulent flows on unstructured, hexahedral grids. GPU-accelerated computations with GALÆXI are possible on GPU hardware by interfacing Fortran source code to the vendor models CUDA C++ for NVIDIA and HIP C++ for AMD. The DGSEM implementation in GALÆXI was verified using the method of manufactured solutions to rigorously confirm the expected order of convergence. Simulations of a compressible Taylor-Green-Vortex also demonstrated excellent agreement with reference solutions across all supported architectures. GALÆXI achieved near ideal strong and weak scaling on GPU hardware from both NVIDIA and AMD. In the largest case, GALÆXI performed a simulation with 67.1 billion degrees of freedom on 65,536 AMD MI250X graphics compute devices with a parallel efficiency of 82.6%. Comparing node-to-node performance, GPU simulations offered speedups between 7.75x and 8.08x over CPU computations in time-to-solution while consuming less than half the energy. To demonstrate GALÆXI's effectiveness for production-scale simulations, wall-resolved large eddy simulations of the transonic flow past a NACA 64A-110 airfoil and an ONERA OAT15A airfoil under shock buffet conditions were computed.

physics.flu-dyn↗

arXiv2026

A Provably Robust Multi-Jet Framework applied to Active Flow Control of an Airfoil in Weakly Compressible Flow

Reinforcement learning has by now become well established in finding excellent flow control strategies for a variety of scenarios. Existing literature has focused on using a simple two-jet solution (and variants there-of) or a straightforward mean-centered multi-jet setup. This mean-centering approach is however non-injective in nature, such that distinct action predictions by the actor network can lead to the same implemented jet-intensities. Thus, the potential of true multi-jet setups still remains unexplored. To this end, in this study we first theoretically analyze multi-jet setups, highlighting the aforementioned pitfall and offer a viable alternative. We also derive upper-bounds on the running costs of these setups, and find the proposed approach to have a jet-count-independent maximum running cost (compared to a near-linear scaling for the traditional setup). The mean-centered and proposed multi-jet setups are applied to a variety of flow-configurations, to test performance and learning capabilities. The new formulation proves effective in learning more complex flow-control strategies, coordinating the jets in a sophisticated manner so as to produce favorable outcomes at minimal actuation cost. For the cylinder-in-channel case, this results in drag and total-force suppression to beyond an idealized symmetric case, whereas for the airfoil the separation region is minimized and significant improvements in aerodynamic efficiency are observed (from 53% up to 73% depending on jet configuration). Additionally, we also incorporate some best practices from traditional RL literature to show fast, reproducible and reliable learning, thereby bringing down the upfront training costs. This study thus provides a robust and mathematically grounded approach to multi-jet design and closes a hitherto overlooked theoretical gap.

physics.flu-dyn↗

arXiv2025

In-Memory Load Balancing for Discontinuous Galerkin Methods on Polytopal Meshes

High-order accurate discontinuous Galerkin (DG) methods have emerged as powerful tools for solving partial differential equations such as the compressible Navier-Stokes equations due to their excellent dispersion-dissipation properties and scalability on modern hardware. The open-source DG framework FLEXI has recently been extended to support DG schemes on general polytopal elements including tetrahedra, prisms, and pyramids. This advancement enables simulations on complex geometries where purely hexahedral meshes are difficult or impossible to generate. However, the use of meshes with heterogeneous element types introduces a workload imbalance, a consequence of the temporal evolution of modal rather than nodal degrees of freedom and the accompanying transformations. In this work, we present a lightweight, system-agnostic in-memory load balancing strategy designed for high-order DG solvers. The method employs high-precision runtime measurements and efficient data redistribution to dynamically reassign mesh elements along a space-filling curve. We demonstrate the effectiveness of the approach through simulations of the Taylor-Green vortex and large-scale parallel runs on the EuroHPC pre-exascale system MareNostrum 5. Results show that the proposed strategy recovers a significant fraction of the lost efficiency on heterogeneous meshes while retaining excellent strong and weak scaling.

math.NA↗

arXiv2025

Harvesting energy consumption on European HPC systems: Sharing Experience from the CEEC project

Energy efficiency has emerged as a central challenge for modern high-performance computing (HPC) systems, where escalating computational demands and architectural complexity have led to significant energy footprints. This paper presents the collective experience of the EuroHPC JU Center of Excellence in Exascale CFD (CEEC) in measuring, analyzing, and optimizing energy consumption across major European HPC systems. We briefly review key methodologies and tools for energy measurement as well as define metrics for reporting results. Through case studies using representative CFD applications (waLBerla, FLEXI/GALÆXI, Neko, and NekRS), we evaluate energy-to-solution and time-to-solution metrics on diverse architectures, including CPU- and GPU-based partitions of LUMI, MareNostrum5, MeluXina, and JUWELS Booster. Our results highlight the advantages of accelerators and mixed-precision techniques for reducing energy consumption while maintaining computational accuracy. Finally, we advocate the need to facilitate energy measurements on HPC systems in order to raise awareness, teach the community, and take actions toward more sustainable exascale computing.

cs.DC↗

arXiv2025

Entropy stable high-order discontinuous Galerkin spectral-element methods on curvilinear, hybrid meshes

Hyperbolic-parabolic partial differential equations are widely used for the modeling of complex, multiscale problems. High-order methods such as the discontinuous Galerkin (DG) scheme are attractive candidates for their numerical approximation. However, high-order methods are prone to instabilities in the presence of underresolved flow features. A popular counter measure to stabilize DG methods is the use of entropy-stable formulations based on summation-by-parts (SBP) operators. The present paper aims to construct a robust and efficient entropy-stable discontinuous Galerkin spectral element method (DGSEM) of arbitrary order on heterogeneous, curvilinear grids composed of triangular and quadrilateral elements or hexahedral, prismatic, tetrahedral and pyramid elements. To the author's knowledge, with the exception of hexahedral and quadrilateral elements, entropy-stable DGSE operators have been constructed exclusively for tetrahedral and triangular meshes. The extension of the DGSEM to more complex element shapes is achieved by means of a collapsed coordinate transformation. Legendre--Gauss quadrature nodes are employed as collocation points in conjunction with a generalized SBP operator and entropy-projected variables. The purely hyperbolic operator is extended to hyperbolic-parabolic problems by the use of a lifting procedure. To circumvent the penalizing time step restriction imposed by the collapsing, modal rather than nodal degrees of freedom are evolved in time, thereby relying on a memory-efficient weight-adjusted approximation to the inverse of the mass matrix. Essential properties of the proposed numerical scheme including free-stream preservation, polynomial and grid convergence as well as entropy conservation / stability are verified. Finally, with the flow around the common research model, the applicability of the presented method to real-world problems is demonstrated.

math.NA↗

arXiv2025

Comparison of Entropy Stable Collocation High-Order DG Methods for Compressible Turbulent Flows

High-order methods are well-suited for the numerical simulation of complex compressible turbulent flows, but require additional stabilization techniques to capture instabilities arising from the underlying non-linear hyperbolic equations. This paper provides a detailed comparison of the effectiveness of entropy stable discontinuous Galerkin methods for the stabilization of compressible (wall-bounded) turbulent flows. For this investigation, an entropy stable discontinuous Galerkin spectral element method is applied on Gauss-Legendre and Gauss-Lobatto nodes. In the compressible regime, an additional stabilization technique for shock capturing based on a convex blending of a low-order finite volume with the high-order discontinuous Galerkin operator is utilized. The present investigation provides a systematic study from convergence tests, to the Taylor-Green vortex and finally to a more intricate turbulent wall-bounded 3D diffuser flow, encompassing both weakly compressible and compressible regimes. The comparison demonstrates that the DGSEM on Gauss-Lobatto nodes is less accurate due to the lower integration accuracy. Conversely, it is faster than the DGSEM on Gauss-Legendre nodes due to a less severe time step restriction and simpler numerical operator. To the author's knowledge, this is the first time for which a comparison of entropy stable DGSEM on Gauss-Lobatto and Gauss-Legendre has been performed for compressible, wall-bounded turbulent flows with separation.

physics.flu-dyn↗

arXiv2025

Entropy stable shock capturing for high-order DGSEM on moving meshes

In this paper, a shock capturing for high-order entropy stable discontinuous Galerkin spectral element methods on moving meshes is proposed using Gauss--Lobatto nodes. The shock capturing is achieved via the convex blending of the high-order scheme with a low-order finite volume subcell operator. The free-stream and convergence properties of the hybrid scheme are demonstrated numerically along with the entropy stability and shock capturing capabilities.

math.NA↗

arXiv2025

Invariant Control Strategies for Active Flow Control using Graph Neural Networks

Reinforcement learning has gained traction for active flow control tasks, with initial applications exploring drag mitigation via flow field augmentation around a two-dimensional cylinder. RL has since been extended to more complex turbulent flows and has shown significant potential in learning complex control strategies. However, such applications remain computationally challenging due to its sample inefficiency and associated simulation costs. This fact is worsened by the lack of generalization capabilities of these trained policy networks, often being implicitly tied to the input configurations of their training conditions. In this work, we propose the use of graph neural networks to address this particular limitation, effectively increasing the range of applicability and getting more value out of the upfront RL training cost. GNNs can naturally process unstructured, threedimensional flow data, preserving spatial relationships without the constraints of a Cartesian grid. Additionally, they incorporate rotational, reflectional, and permutation invariance into the learned control policies, thus improving generalization and thereby removing the shortcomings of commonly used CNN or MLP architectures. To demonstrate the effectiveness of this approach, we revisit the well-established two-dimensional cylinder benchmark problem for active flow control. The RL training is implemented using Relexi, a high-performance RL framework, with flow simulations conducted in parallel using the high-order discontinuous Galerkin framework FLEXI. Our results show that GNN-based control policies achieve comparable performance to existing methods while benefiting from improved generalization properties. This work establishes GNNs as a promising architecture for RL-based flow control and highlights the capabilities of Relexi and FLEXI for large-scale RL applications in fluid dynamics.

cs.LG↗

arXiv2025

Efficient computation of particle-fluid and particle-particle interactions in compressible flow

Particle collisions are the primary mechanism of inter-particle momentum and energy exchange for dense particle-laden flow. Accurate approximation of this collision operator in four-way coupled Euler-Lagrange approaches remains challenging due to the associated computational cost. Adopting a deterministic collision model and a hard-sphere approach eases time step constraints but imposes non-locality on distributed memory architectures, necessitating the inclusion of collision partners from each grid element in the vicinity. Retaining high-order accuracy and parallel efficiency also ties into the correct and compact treatment of the particle-fluid coupling, where adequate kernels are required to effectively project the work of the particles to the Eulerian grid. In this work, we present an efficient particle collision and projection operator based on an MPI+MPI hybrid approach to enable time-resolved and high-order accurate simulations of compressible, four-way coupled particle-laden flows at dense concentrations. A distinct feature of the proposed particle collision algorithm is the efficient calculation of exact binary inter-particle collisions on arbitrary core counts. Combining the particle operator with a hybrid discretization operator based on a high-order discontinuous Galerkin method and a localized low-order finite volume operator allows an accurate treatment of highly compressible particle-laden flows. The approach is extensively validated against a range of benchmark problems. Contrary to literature, the scaling properties are demonstrated on state-of-the-art high performance computing systems. Finally, the proposed algorithm is compatible with unstructured, curved high-order grids which permits the handling of complex geometries as is emphasized by application of the framework to large-scale application cases.

physics.comp-ph↗

arXiv2024

GALÆXI: Solving complex compressible flows with high-order discontinuous Galerkin methods on accelerator-based systems

This work presents GALAEXI as a novel, energy-efficient flow solver for the simulation of compressible flows on unstructured meshes leveraging the parallel computing power of modern Graphics Processing Units (GPUs). GALAEXI implements the high-order Discontinuous Galerkin Spectral Element Method (DGSEM) using shock capturing with a finite-volume subcell approach to ensure the stability of the high-order scheme near shocks. This work provides details on the general code design, the parallelization strategy, and the implementation approach for the compute kernels with a focus on the element local mappings between volume and surface data due to the unstructured mesh. GALAEXI exhibits excellent strong scaling properties up to 1024 GPUs if each GPU is assigned a minimum of one million degrees of freedom degrees of freedom. To verify its implementation, a convergence study is performed that recovers the theoretical order of convergence of the implemented numerical schemes. Moreover, the solver is validated using both the incompressible and compressible formulation of the Taylor-Green-Vortex at a Mach number of 0.1 and 1.25, respectively. A mesh convergence study shows that the results converge to the high-fidelity reference solution and that the results match the original CPU implementation. Finally, GALAEXI is applied to a large-scale wall-resolved large eddy simulation of a linear cascade of the NASA Rotor 37. Here, the supersonic region and shocks at the leading edge are captured accurately and robustly by the implemented shock-capturing approach. It is demonstrated that GALAEXI requires less than half of the energy to carry out this simulation in comparison to the reference CPU implementation. This renders GALAEXI as a potent tool for accurate and efficient simulations of compressible flows in the realm of exascale computing and the associated new HPC architectures.

cs.MS↗

arXiv2023

Towards Exascale CFD Simulations Using the Discontinuous Galerkin Solver FLEXI

Modern high-order discretizations bear considerable potential for the exascale era due to their high fidelity and the high, local computational load that allows for computational efficiency in massively parallel simulations. To this end, the discontinuous Galerkin (DG) framework FLEXI was selected to demonstrate exascale readiness within the Center of Excellence for Exascale CFD (CEEC) by simulating shock buffet on a three-dimensional wing segment at transsonic flight conditions. This paper summarizes the recent progress made to enable the simulation of this challenging exascale problem. For this, it is first demonstrated that FLEXI scales excellently to over 500 000 CPU cores on HAWK at the HLRS. To tackle the considerable resolution requirements near the wall, a novel wall model is proposed that takes compressibility effects into account and yields decent results for the simulation of a NACA 64A-110 airfoil. To address the shocks in the domain, a finite-volume-based shock capturing method was implemented in FLEXI, which is validated here using the simulation of a linear compressor cascade at supersonic flow conditions, where the method is demonstrated to yield efficient, robust and accurate results. Lastly, we present the TensorFlow-Fortran-Binding (TFFB) as an easy-to-use library to deploy trained machine learning models in Fortran solvers such as FLEXI.

cs.DC↗

arXiv2022

A framework for high-fidelity particle tracking on massively parallel systems

Particle-laden flows occur in a wide range of disciplines, from atmospheric flows to renewable energy to turbomachinery. They generally pose a challenging environment for the numerical prediction of particle-induced phenomena due to their often complex geometry and highly instationary flow field which covers a wide range of spatial and temporal scales. At the same time, confidence in the evolution of the particulate phase is crucial for the reliable prediction of non-linear effects such as erosion and fouling. As a result, the multiscale nature requires the time-accurate integration of the flow field and the dispersed phase, especially in the presence of transition and separation. In this work, we present the extension of the open-source high-order accurate CFD framework FLEXI towards particle-laden flows. FLEXI is a massively parallel solver for the compressible Navier-Stokes-Fourier equations which operates on (un-)structured grids including curved elements and hanging nodes. An efficient particle tracking approach in physical space based on methods from ray-tracing is employed to handle intersections with curved boundaries. We describe the models for a one- and two-way coupled dispersed phase and their numerical treatment, where particular emphasis is placed on discussing the background and motivation leading to specific implementation choices. Special care is taken to retain the excellent scaling properties of FLEXI on high performance computing infrastructures during the complete tool chain including high-order accurate post-processing. Finally, we demonstrate the applicability of the extended framework to large-scale problems.

physics.comp-ph↗

arXiv2020

A Neural Network based Shock Detection and Localization Approach for Discontinuous Galerkin Methods

The stable and accurate approximation of discontinuities such as shocks on a finite computational mesh is a challenging task. Detection of shocks or strong discontinuities in the flow solution is typically achieved through a priori troubled cell indicators, which guide the subsequent action of an appropriate shock capturing mechanism. Arriving at a stable and accurate solution often requires empirically based parameter tuning and adjustments of the indicator settings to the discretization and solution at hand. In this work, we propose to separate the task of shock detection and shock capturing more strongly and aim to develop a shock indicator that is robust, accurate, requires minimal user input and is suitable for high order element-based methods like discontinuous Galerkin and flux reconstruction methods. The novel indicator is learned from analytical data through a supervised learning strategy; its input is given by the high order solution field, its output is an element-local map of the shock position. We use state of the art methods from edge detection in image analysis based on deep convolutional multiscale networks and deep supervision to train the indicators. The resulting networks are then used as black box indicators, showing their robustness and accuracy on well established canonical testcases. All simulations are run ab initio using the developed indicators, showing that they provide also stability during the strongly transient phases. In particular for high order schemes with large cells and considerable inner-cell resolution capabilities, we demonstrate how the additional accurate prediction of the position of the shock front can be exploited to guide inner-element shock capturing strategies.

math.NA↗