This paper presents an efficient hybrid discontinuous Galerkin (HDG) scheme for solving a class of nonlinear and highly anisotropic elliptic problems, characterized by a diffusion tensor where the diffusivity on a given direction scales with the anisotropy ratio $1/\varepsilon$ ($\varepsilon\ll1$ is a positive constant) and is a function of the solution. Motivated by magnetized plasma simulations in tokamaks, these problems often involve anisotropy ratios as high as $10^{10}$ and a diffusivity vanishing on the anisotropy direction with the solution. The present work extends to nonlinear problems the quasi-asymptotic preserving (QAP) HDG scheme we very recently introduced in Mehrenberger et al., 2025 and which has shown encouraging suppressing the spurious effects of anisotropy. The proposed HDG-QAP scheme utilizes a linearization that combines a modified Uzawa iteration with the fixed-point iteration. Numerical results based on manufactured solutions demonstrate the robustness of the scheme, showing high-order accuracy and remarkable stability even in the limit case $\varepsilon=0$.

This work introduces a novel flux-surface-aligned algorithm to extend the state-of-the-art SOLEDGE3X code to non-axisymmetric magnetic configurations, and thereby addresses the growing demand for comprehensive plasma edge simulations in both tokamak and stellarator geometries. Such configurations pose significant challenges, as toroidal distortions of magnetic surfaces invalidate conventional mesh construction strategies that rely on flux-aligned meshes to handle the strong anisotropy between parallel and perpendicular transport. The proposed method transforms the existing 2D mesh into a 3D flux-aligned mesh by computing a 3D poloidal flux function, $\Psi^{3D}$, derived from vacuum coil perturbations. To accommodate the new geometry, a three-dimensional curvilinear metric formulation is introduced, where metric coefficients are evaluated at cell centers using finite-difference calculations based on the 3D mesh corner positions and are consistently integrated into all discrete operators. Verification tests using manufactured solutions confirm that the new algorithm preserves the accuracy and the robustness of the original axisymmetric version. In particular, the resulting flux surfaces are demonstrated to remain aligned with the magnetic field to first order in the perturbation amplitude. The method of manufactured solutions (MMS) also shows the second-order spatial accuracy of the transport operators. The enhanced capabilities of the solver are demonstrated through transport simulations of toroidal field ripple effects in the WEST tokamak, where heat flux patterns are consistent with experimental observations, and strong plasma asymmetries emerge in detached divertor regimes. Additionally, a proof-of-concept simulation on a stellarator-like geometry validates the algorithm’s ability to consistently model plasma transport, neutral dynamics, limiter boundary conditions, and density-control feedback in fully three-dimensional configurations with nested flux surfaces.

This fundamental research investigates the proof-of-concept effect of a counterflow Shell-and-Spiral Coil Heat Exchanger (SSCHE) on CO emissions from a B30-fueled single-cylinder 7 HP diesel engine under no-load conditions, without dynamometer loading, and establishes a scientific basis before advancing to prototype development with bypass valve temperature control. CFD simulation using SolidWorks Flow Simulation 2023 predicted fuel outlet temperatures of 55 degrees C, 78 degrees C, and 92 degrees C for engine speeds of 1000, 1250, and 1500 rpm, respectively. The experimental setup on a B30-fueled Jiang FA R175 A diesel engine demonstrated actual fuel outlet temperatures of 40.14 +/- 5.77 degrees C, 56.18 +/- 18.26 degrees C, and 77.34 +/- 7.01 degrees C, with CFD deviations of 27.0%, 28.0%, and 15.9%, respectively. CO emission analysis demonstrated significant reductions: 56.03% at 1000 rpm, 27.98% at 1000 and 1250 rpm, respectively, but showed a 7.79% increase at 1500 rpm. Findings reveal the best CO reduction was observed at fuel outlet temperatures of 40 degrees C-56 degrees C (low to medium rpm) under the no-load conditions tested. The three-point dataset is insufficient to establish a temperature optimum, and future controlled experiments using bypass valve modulation are required to achieve this. Statistical analysis: Cohen's d = 3.12, 95% CI [96.8-114.4] ppm, p < 0.001 at 1000 rpm (very large effect); Cohen's d = 2.10, p < 0.001 at 1250 rpm (large effect); and significant increase in CO at 1500 rpm (d = 0.37, p = 0.003). CFD deviations (15.9%-28.0%) attributed to specification-based boundary conditions and steady-state assumptions; the model is treated as a preliminary design tool throughout the study.

This paper presents a joint experimental and numerical investigation into the physics of long limited plasma ramp-up in tokamaks with tungsten (W) first walls, a critical phase for ITER operation. The comparison between the average plasma quantities simulated using the SolEdge-HDG code and the measurements taken during three successive WEST discharges after boronisation shows how challenging it is to predict this phase. While simulations reproduce the general trends at the midplane, with reasonable match in density profiles, they consistently underestimate core temperatures possibly due to too large perpendicular heat conductivity. On the contrary, at the high field side (HFS) limiter, simulations overestimate the measured quantities, and highlights the limitation of using Bohm boundary conditions at grazing magnetic angles. Experimental measurements reveal that the boron layer is rapidly eroded, on a timescale comparable to a single ITER discharge. The subsequent transition from a boron-coated to a tungsten wall increases recycling and significantly degrades the core plasma, reducing the electron temperature by nearly half due to W contamination, despite wall parameters remaining stable. Furthermore, comparisons with Langmuir probes, bolometry, reflectometry, and spectroscopy indicate that the experimental far scrape-Off layer (SOL) is significantly wider than simulated. This wide SOL implies that boron erosion extends along the entire HFS limiter rather than being confined to the contact point. This work highlights some characteristics of the plasma during this phase of the discharge and emphasizes the current modelling issues that need to be resolved in order to obtain reliable predictions concerning the ITER ramp-up.

This paper proposes a quasi-asymptotic-preserving hybrid discontinuous Galerkin (HDG-QAP) scheme for the resolution of highly anisotropic diffusion problems. The HDG-QAP scheme introduces an auxiliary unknown that serves to capture the information on the dominant diffusion scale. We show that it is well-posed for any ε > 0, with ε being a small constant and 1/ ε characterizing the anisotropy strength, and that its solution is bounded uniformly in ε . At this point, the standard HDG procedure, namely, the static condensation on the numerical trace, turns out to violate these uniform bounds. Instead, we show that the static condensation on the auxiliary unknown does lead to similar bounds uniform in ε , but its resolution is costly in terms of computational effort. Therefore, we propose a relaxation method, called the HDG-QAP Uzawa iteration, to overcome this challenge, in which each iteration is fast to compute. We show that the HDG-QAP Uzawa iteration converges for any ε > 0, but also for ε = 0. Finally, we provide some numerical examples to confirm the findings, and in particular to show that the proposed HDG-QAP Uzawa iteration works well even with a severe anisotropy ε = 10 −15 where the quality of the numerical solutions is unaffected by the anisotropy strength.

This work presents a conservative treatment of solid boundaries that enforces strict mass conservation and mass-energy flux consistency at solid boundaries while maintaining the efficiency of the hybrid Lattice–Boltzmann/Finite–Volume (LB/FV) method for simulating compressible fluid flows. In this method, the mass and momentum equations are solved using the hybrid recursive regularized LB (HRR) algorithm, while the total energy equation is solved using a FV scheme. The strategy proposed by Zhao et al. (2020) is used to strongly couple LB mass and FV energy fluxes by evaluating the latter directly from the distribution functions of the LBM kinetic part. Furthermore, a modified regularized boundary condition is introduced to reconstruct the pre-collision populations at the boundaries. At boundary nodes, the density is updated from a lattice-consistent continuity balance, in which the mass variation of the boundary control volume is computed from the post-collision population exchange along the available fluid links, while zero mass exchange is imposed across blocked solid links. A series of validation cases involving conservation tests, thermal wall conditions, wall-force evaluation, and compressible-flow benchmarks demonstrates the effectiveness of the proposed boundary treatment in restoring the targeted discrete conservation properties without degrading the prediction of relevant wall and aerodynamic quantities.

The highly anisotropic, multi-scale nature of fusion plasma simulations in tokamaks, combined with the complexity in the geometries of plasma-facing components and magnetic equilibrium, challenges numerical schemes. They therefore require the development of advanced numerical techniques to enhance computational efficiency and enable codes to simulate realistic plasma configurations relevant to tokamak operation. This paper proposes an adaptive mesh refinement strategy (-adaptivity) in the SolEdge-HDG code for the resolution of 2D fluid-drift Braginskii equations using the Hybrid Discontinuous Galerkin (HDG) method. The strategy is based on an oscillation indicator implemented to detect under-resolved regions and dynamically refine the mesh, associated with an a posteriori accuracy indicator built on the local difference between the solution at order and the post-processed one at order considered as reference. The method thus enables both refinement, where necessary, as well as coarsening in regions where the solution is smooth. Numerical results obtained with this method in realistic tokamak geometry and plasma conditions show significant reductions in computational resources and an improvement in code robustness while maintaining high accuracy, particularly in regions with steep gradients or near the sharp angles of the tokamak walls. This work highlights the potential of such -adaptivity technique to optimize transport simulations in realistic tokamak configurations, offering a fully automated, goal-oriented mesh refinement strategy. In addition to optimizing the numerical cost of simulation, this strategy offers all users a fully automated means of designing a mesh in any tokamak geometry.

Even small fluctuations in the magnetic field are known to impact edge plasma turbulence and transport properties in magnetic confinement fusion devices. Magnetic induction modifies the parallel electric field, and as such, it impacts the parallel current in Ohm's law. In addition, magnetic fluctuations can induce geometrical and topological changes in the magnetic field structure, leading to parallel transport across the equilibrium magnetic surfaces. This paper presents the new drift-reduced fluid electromagnetic model implemented in SOLEDGE3X [Bufferand et al., Nuc. Fus. 2021]. Based on a domain decomposition, a specific numerical scheme is proposed using conservative second-order finite volumes associated with a semi-implicit time advancement. The coupling between the parallel current jll and the parallel electromagnetic potential All in Ohm's and Ampere's laws is treated using a new toroidally and poloidally staggered grid. While adding All doubles the size of the vorticity operator to be inverted, numerical tests show that the inclusion of the finite mass of the electrons in the new model acts as an upper limit on the parallel diffusion coefficient, thus improving the conditioning of the matrix at high plasma temperatures when the parallel resistivity rill approaches zero. The changes in the algorithm are verified using the method of manufactured solutions. The implementation is first validated with respect to theoretical linear stability results, recovering the transition from Alfv & eacute;n to thermal electron waves as the perpendicular wavenumber increased. Comparisons with available results in the literature show a qualitative agreement on a single blob propagation in a limited slab geometry. The first 3D simulations of plasma edge turbulence in TCV demonstrate the capability of the new solver to handle realistic tokamak configurations. The explicit-implicit time integration scheme enables one to compare electrostatic and electromagnetic effects using the same solver, with or without electron inertia and magnetic flutter. This ability opens the way to a better understanding of the impact of the electrostatic and electromagnetic mechanisms on the transport and turbulence properties in realistic tokamak configurations.

A steady-state, 1D semi-analytical model for prompt redeposition based on the separation between redeposition caused by the electric field in the sheath and redeposition related to gyromotion is here described. The model allows for the estimation of not only the fraction of promptly redeposited flux but also the energy and angular distribution of the non-promptly redeposited population, along with their average charge state. Thus, the temperature and mean parallel-to-B velocity of the non-promptly redeposited flux are also available. The semi-analytical model was validated against equivalent Monte Carlo simulations across a broad range of input parameters. In this paper the eroded material under exam was tungsten (W) for which the code demonstrated consistent agreement with respect to numerical results, within its defined validity limits. The model can theoretically provide a solution for any material, temperature and electron density profile in the sheath, monotonic potential drop profile, and sputtered particles energy and angular distribution at the wall. As such, this code emerges as a potential tool for addressing the boundary redeposition phenomenon in fluid impurity transport simulations.

First principle modelling of edge plasma turbulence including neutrals and plasma recycling on the wall remains a challenge, in particular due to the long time scales necessary to simulate to reach particle balance. In this contribution, we propose a strategy to address these long time scales with the fluid code SOLEDGE, resorting to 2D reduced models for turbulence as well as 3D coarse grid simulations. The approach is applied to simulate TCV-X21 reference plasma scenario for edge turbulence modelling validation.

Plasma-wall interaction is one of the key research topics on the way to controlled fusion. To study the best operational designs with reduced heat and particle fluxes onto tokamak plasma facing components (PFCs) comprehensive plasma simulations are required. A recent implementation of a hybridized discontinuous Galerkin scheme into a new version of SolEdge code has the advantage of using magnetic equilibrium-free mesh. This allows us to conduct pioneering 2-D transport simulations of a full discharge in the WEST tokamak. In this work, we implemented plasma transport coefficients as functions of coordinate in the poloidal plane and neutral diffusion as a function of neutral mean free path. Moreover, the perpendicular convection flux terms were added to the code. Using the new features, a few test cases were investigated. The influence of nonconstant transport coefficients on the simulated particle and heat fluxes onto the WEST tokamak PFCs are demonstrated.

The mission of WEST (tungsten-W Environment in Steady-state Tokamak) is to explore long pulse operation in a full tungsten (W) environment for preparing next-step fusion devices (ITER and DEMO) with a focus on testing the ITER actively cooled W divertor in tokamak conditions. Following the successful completion of phase 1 (2016-2021), phase 2 started in December 2022 with the lower divertor made entirely of actively cooled ITER-grade tungsten mono-blocks. A boronization prior the first plasma attempt allowed for a smooth startup with the new divertor. Despite the reduced operating window due to tungsten, rapid progress has been made in long pulse operation, resulting in discharges with a pulse length of 100 s and an injected energy of around 300 MJ per discharge. Plasma startup studies were carried out with equatorial boron nitride limiters to compare them with tungsten limiters, while Ion Cyclotron Resonance Heating assisted startup was attempted. High fluence operation in attached regime, which was the main thrust of the first campaigns, already showed the progressive build up of deposits and appearance of dust, impacting the plasma operation as the plasma fluence increased. In total, the cumulated injected energy during the first campaigns reached 43 GJ and the cumulated plasma time exceeded 5 h. Demonstration of controlled X-Point Radiator regime is also reported, opening a promising route for investigating plasma exhaust and plasma-wall interaction issues in more detached regime. This paper summarises the lessons learned from the manufacturing and the first operation of the ITER-grade divertor, describing the progress achieved in optimising operation in a full W environment with a focus on long pulse operation and plasma wall interaction.

Relaxations of localized over-density in a plane transverse to the magnetic field are numericallyinvestigated under the effect of drift-wave and interchange drives in SOL conditions. Such a controlleddeparture from thermodynamic equilibrium allows the investigation of fundamental processesat play in cross-field transport. Interchange instabilities generate ballistic outward radial flux withlow amplitude zonal flow patterns, whereas drift-wave instabilities result in symmetric radial fluxwith large amplitude zonal flow patterns. When both instabilities are considered, the combinedeffects tend to favor drift-waves, leading to a weaker outward flux with larger zonal flow patterns.

The fluid-drift code SOLEDGE3X, developed by CEA/IRFM in collaboration with Aix-Marseille University, is a powerful tool for simulating transport and turbulence in tokamak edge plasmas with axisymmetric magnetic configurations. In tokamaks such as WEST, the pronounced toroidal magnetic ripple significantly affects plasma confinement and power exhaust, modulating both the poloidal and toroidal components of the equilibrium field. Using a discrete Biot–Savart law, the ripple field is calculated as a magnetic perturbation on the SOLEDGE3X mesh. The transport model and parallel gradient solvers have been enhanced to incorporate the new radial magnetic field component. Preliminary simulations of a WEST scenario reveal a heat deposition pattern in the divertor region consistent with observations from infrared camera experiments.

In the pedestal region, electromagnetic effects affect the evolution of micro-instabilities and plasma turbulence. The transport code Soledge3X developed by the CEA offers an efficient framework for turbulent 3D simulation on an electrostatic model with a fixed magnetic field. The physical accuracy of the model is improved with electromagnetic induction, driven by the local value of the parallel component of the electromagnetic vector potential A parallel to$$ {A}_{\parallel } $$, known from Ampere's law. It is solved implicitly in a coupled system with the vorticity equation on the electric potential phi$$ \Phi $$. The consequence is a basic electromagnetic behavior in the form of shear Alfven waves. A finite electron mass prevents unphysical speeds but requires solving for the time evolution of the parallel current density j parallel to$$ {j}_{\parallel } $$ in the generalized Ohm's law. This term can be analytically included with little computational overhead in the system on phi$$ \Phi $$ and A parallel to$$ {A}_{\parallel } $$ and improves its numerical condition, facilitating the iterative solving procedure. Simulations on a periodic slab case let us observe the predicted bifurcation of the wave propagation speed between the Alfven wave and the electron thermal wave speeds for varying perpendicular wavenumbers. The first results on a circular geometry with a limiter attest to the feasibility of turbulent electromagnetic scenarios.

With ITER, the largest tokamak ever built, and the growing number of fusion energy startups in the world, the need for numerical simulations has never been more crucial to progress towards the successful operation of fusion reactors. From fundamental plasma physics to engineering, a hierarchy of models exists from high-fidelity (gyro-)kinetic models in (5D) 6D to 0D fluid transport models. In this paper, we review the state-of-the-art of 3D turbulence fluid simulations in edge tokamak configurations. The widely used drift-reduced Braginskii equations are introduced together with the dedicated boundary conditions modelling plasma wall interactions. If until recently most of the models were focused on electrostatic turbulence driven by interchange-like instabilities, in recent years models have incorporated electromagnetic effects allowing fluctuations of the magnetic field. Specific features of the edge plasma configurations, which make these equations specially challenging to resolve and stressful for the numerical methods, are detailed. In particular, the strong anisotropy of the flow as well as the complex geometric characteristics lead to the development of dedicated discretization schemes and meshing, which are implemented in state-of-the-art codes reviewed here. It appears that the latter can be differentiated by their mesh construction as well by the manner in which they handle parallel gradients (aligned or not along the magnetic field). The review shows that no consensus on the optimal combination between meshing and discretization schemes, if it exists, has been found. Finally, examples of recent achievements show that 3D turbulence simulations of medium-sized tokamaks are currently achievable, but that ITER-size tokamaks and thermonuclear plasmas still require significant progress.

Boundary plasma simulations are essential to estimate expected divertor and first wall (FW) heat and particle loads on ITER during burning plasma operation. A key missing feature of existing SOLPS simulations (Pitts et al., 2019) is the absence of a plasma solution out to the main chamber walls, essential to self-consistently estimate the gross sputtering of wall material. Here, SOLEDGE3X is applied for the first time to obtain up-to-the wall burning plasma solutions of the ITER boundary plasma at the nominal PSOL = 100 MW of the main SOLPS database simulations, including He ash, Ne seeding but without fluid drifts. Compared with the most recent SOLPS-ITER simulations, our simulations show differences in the exact impurity distribution, but the key results for divertor and wall heat flux remain consistent. In the context of the ITER re-baselining exercise (Pitts, 2024), in which the Be FW armour is proposed to be exchanged for tungsten (W), estimates of W wall sources are key to the assessment of likely core contamination and hence impact on fusion gain. We compare the W gross erosion rates due to the different species excluding W self-sputtering. For the cases simulated spanning 0.27%–0.47% separatrix-averaged Ne concentration and 7.5×1022s−1−1.95×1023s−1 D fuelling, Ne8+ remains the largest contributor to the sputtering flux with the largest source being the outer divertor and baffle. The species-wise contribution to W sputtering changes with fuelling with sputtering due to lower Ne charge states being significant at low D fuelling. In general, the gross W sputtering source is found to decrease with increase in D fuelling and increase with increased Ne seeding.

This work explores the Limiter–Divertor transition (L–D) during the current ramp-up of ITER’s Q = 10 baseline plasma scenario at various central line-integrated density nli values. The analysis, based on transport simulations performed with the latest version of SoleEdge-HDG, focuses on the time evolution of heat and ion particle fluxes, revealing regions of elevated temperature on the inner wall and plasma-facing components (PFCs) despite moderate loads. The investigation also delves into the effects of perpendicular convection flux terms on density build-up, comparing different formulations and their interplay with auxiliary heating sources. Furthermore, the paper shows the impact of taking into account the evolution of the parallel neutral momentum on plasma and neutral density at the targets in the context of an ITER steady-state scenario.

This paper addresses the simulation of internal high-speed turbulent compressible flows using lattice Boltzmann method (LBM) when it is coupled with the immersed boundary method for non-body-fitted meshes. The focus is made here on the mass leakage issue. The recent LBM pressure-based algorithm [Farag et al. Phys. Fluids 32, 066106 (2020)] has shown its superiority on classical density-based algorithm to simulate high-speed compressible flows. Following our previous theoretical work on incompressible flows [Xu et al. Phys. Fluids 34, 065113 (2022)], we propose an averaged mass correction technique to mitigate mass leakage when simulating high-Mach-number compressible flows. It is adapted to deal here with a density, which is decoupled from the zero-moment definition. The simulations focus on two generic but canonical configurations of more complex industrial devices, the straight channel at different angles of inclination at Mach numbers (Ma) ranging from 0.2 to 0.8, and the National Aeronautics and Space Administration Glenn S-duct at Ma = 0.6. The present results show that mass leakage can be a critical issue for the accuracy of the solution and that the proposed correction technique effectively mitigates it and leads to significant improvements in the prediction of the solution.