Solving the radiation transport equation is a challenging task, due to the high dimensionality of the solution's phase space. The commonly used discrete ordinates (S N ) method suffers from ray effects which result from a break in rotational symmetry from the finite set of directions chosen by S N . The spherical harmonics (P N ) equations, on the other hand, preserve rotational symmetry, but can produce negative particle densities. The discrete ordinates (S N ) method, in turn, by construction ensures non-negative particle densities.In this paper we present a modified version of the S N method, the rotated S N (rS N ) method. Compared to S N , we add a rotation and interpolation step for the angular quadrature points and the respective function values after every time step. Thereby, the number of directions on which the solution evolves is effectively increased and ray effects are mitigated. Solution values on rotated ordinates are computed by an interpolation step. Implementation details are provided and in our experiments the rotation and interpolation step only adds 5% to 10% to the runtime of the S N method. We apply the rS N method to the line-source and a lattice test case, both being prone to ray effects. Ray effects are reduced significantly, even for small numbers of quadrature points. The rS N method yields qualitatively similar solutions to the S N method with less than a third of the number of quadrature points, both for the line-source and the lattice problem. The code used to produce our results is freely available and can be downloaded [4].

Our global understanding of the power exhaust in tokamaks, and its implications for both steady‐state and transient heat loads on divertor and limiter PFCs, is still poor. In transient situations in particular, such as during start‐up or control operations, the evolution of particles and heat fluxes is little known, although they are critical for the safety of the machine. The heat load is largely determined by the physics of the Scrape‐Off Layer (SOL), and therefore, it depends to a large extent on the geometry of the magnetic surfaces as well as on the geometry of wall components. A better characterization of the heat exhaust mechanisms is therefore required to improve the capabilities of the transport codes in terms of geometrical description of the wall components and in terms of the description of the magnetic geometry. For transient simulations, it becomes crucial to be able to deal with non‐stationary magnetic configurations. In particular, avoiding expensive re‐meshing of the computational domain is mandatory. In an attempt to achieve these goals, we propose a new fluid solver based on a high‐order hybrid discontinuous Galerkin (HDG) finite element method. Capitalizing on the experience acquired in the development of the SOLEDGE2D‐EIRENE transport model, we propose to study edge plasma transport in the frame of a reduced model (but containing most of the challenging issues regarding accurate numerical simulations) based on electron density and parallel momentum. The code is verified using manufactured solutions and validated using well‐referenced simulations in a realistic WEST geometry. Finally, we demonstrate how the particle fluxes at the wall vary in our model when the magnetic equilibrium evolves in time, particularly during the equilibrium construction skip from a limiter configuration to a diverted one at the beginning of the operation.

No abstract

Sweeping is a commonly used procedure to explicitly solve the discrete ordinates equation, which itself is a common approximation of the neutron transport equation. To sweep through the computational domain, an ordering of the spatial cells is required that obeys the flow of information. We show that this ordering can always be found, assuming a discretization of the spatial domain with regular triangles with no hanging nodes.

ZusammenfassungEin Anspruch des mathematischen Modellierungsunterrichts in der Schule sollte es sein, besonders aktuelle Probleme und interessante neue Technologien aus dem Alltag der Schüler/innen einzubeziehen. Dies gilt insbesondere, wenn sie eine didaktische Reduktion auf elementare (schul-)mathematische Inhalte leicht zulassen. Künstliche Intelligenz (KI) zieht sich durch verschiedene Bereiche von Wissenschaft und Technik und verbirgt sich insbesondere hinter zahlreichen Anwendungen unseres Alltags.In diesem Beitrag wird diskutiert, wie ein zeitgemäßer Mathematikunterricht durch die Modellierung realer, schülernaher Probleme aus dem Bereich KI bereichert werden kann. Dazu werden zwei Methoden und deren didaktische Reduktion für den Einsatz in einem computergestützten Mathematikunterricht vorgestellt.Bei der problemorientierten Diskussion beider Methoden werden zwei alltägliche Problemstellungen in den Blick genommen: Zum einen Klassifizierungsprobleme und deren Lösung mithilfe der sogenannten Stützvektormethode (SVM), die auf der Berechnung des Abstandes von Punkten zu Hyperebenen beruht; zum anderen Empfehlungssysteme, die auf einer Matrix-Faktorisierung basieren können.Zu beiden Problemstellungen wurden digitale Lernmaterialien für Oberstufenschüler/innen entwickelt, die im Rahmen von eintägigen Workshops zur mathematischen Modellierung bereits mehrfach erprobt wurden. Die digitale Umsetzung als Jupyter Notebooks wird abschließend beschrieben und steht den Leser/innen als Open Educational Resources unter einer Creative Commons Lizenz zur Verfügung.