We develop structure-preserving methods for the compressible Euler equations with gravity in vector-invariant form and potential temperature as a prognostic variable within the flux-differencing discontinuous Galerkin spectral element method (DGSEM) framework. By discretizing the nonconservative terms as symmetric and antisymmetric products, we derive two-point numerical fluxes that conserve both the thermodynamic entropy and the total energy. Moreover, we design an entropy-stable numerical flux that is well-balanced for both isothermal and isentropic background states. All properties are shown to carry over to the high-order DGSEM on general curvilinear meshes. Several numerical examples confirm the theoretical findings and show the robustness and accuracy of the scheme for use in modern dynamical cores for atmospheric flows.
The reproducibility repository is available on GitHub and Zenodo.
]]>]]>Computational Science and Engineering (CSE) combines expertise at the intersection of engineering, applied mathematics, and computer science to form powerful methods for model-based design and model-based decision support across disciplines. Today, CSE methods and tools have an impact as an enabling technology in the development of increasingly sustainable products, processes, and operations. However, sustainability is rarely considered holistically in the context of CSE itself. This paper develops a perspective on sustainability in CSE by distinguishing CSE as an enabler of sustainability in application domains from sustainability within CSE itself, which rests on two complementary pillars: sustainable computing and sustainable software. We present illustrative examples for these pillars and derive recommendations and best practices for CSE stakeholders. Developing an understanding of how sustainability can create added value in CSE is an important future direction for the field. This calls for a concerted effort within the CSE community to define measurable outcomes and best practices that embed sustainability as a core design principle rather than an afterthought.
]]>Potpourri der Mathematik und Informatik
Wie spannend Mathematik und Informatik sein können, zeigt der interaktive Mitmach-Stand der beiden Institute für Mathematik und Informatik der JGU beim Wissenschaftsmarkt! Ihr seid eingeladen in faszinierende Themenwelten einzutauchen – zum Ausprobieren, Staunen und Verstehen. Gemeinsam entdecken wir die Geometrie von Seifenhäuten, basteln und adoptieren Polyeder, bauen Platonische Körper und Fraktale, lösen spielerisch knifflige Pack- und Färbeprobleme und simulieren physikalische Phänomene am Computer. Eine spannende Reise durch die Welt der Zahlen, Formen und Algorithmen – praxisnah, kreativ und mit Aha-Effekt! Mathematik und Informatik live erleben!
Spätestens nach den Rekordtemperaturen im Juni 2026 in Mainz und der dazugehörigen Berichterstattung, dass solche Hitzewellen vor einigen Jahren noch unmöglich gewesen wären, ist das Thema Klimawandel in aller Munde. Um den Klimawandel und seine Auswirkungen zu verstehen und sachlich diskutieren zu können, werden Modelle und Simulationen benötigt. In diesem Kurs werden wir einen kleinen Blick hinter die Kulissen einfacher Klimamodelle werfen. Dazu betrachten wir zunächst die notwendigen Grundlagen der Analysis (Ableitungen und verwandte Begriffe) und nutzen diese für eine mathematisch-physikalische Modellierung des Klimas. Um die entstehenden (Differential-) Gleichungen zu lösen, verwenden wir numerische Verfahren, die wir gemeinsam programmieren werden.
You can find more information on the official website. Some lecture notes will be provided on site. You can find additional (programming/code) material on https://github.com/ranocha/2026_MMA_Klima.
]]>We propose and study a class of arbitrarily high-order numerical discretizations that preserve multiple invariants and are essentially explicit (they do not require the solution of any large systems of algebraic equations). In space, we use Fourier Galerkin methods, while in time we use a combination of orthogonal projection and relaxation. We prove and numerically demonstrate the conservation properties of the method by applying it to the Benjamin–Bona–Mahony, Korteweg–de Vries, and nonlinear Schrödinger (NLS) PDEs as well as a hyperbolic approximation of NLS. For each of these equations, the proposed schemes conserve mass, momentum, and energy up to numerical precision. We show that this conservation leads to reduced growth of numerical errors for long-term simulations.
The reproducibility repository is available on GitHub and Zenodo.
As usual, you can find the preprint on arXiv.
]]>Structure-preserving discontinuous Galerkin (DG) methods typically improve the robustness of high order simulations of real fluids. In addition to conservation, key structures include the preservation of pressure equilibrium and satisfaction of at least one entropy inequality. In this work, we investigate conservative discretizations using exactly pressure equilibrium conserving (EPEC) and approximately pressure equilibrium conserving (APEC) flux differencing DG formulations, as well as entropy stable formulations through the use of minimally dissipative corrections for non-ideal equations of state (EOS). We introduce an analysis of EPEC schemes and a new procedure for designing such fluxes based on a generalization of Tadmor’s shuffle condition. We also analyze APEC DG schemes and show that the incorporation of dissipative interface penalization terms does not significantly increase pressure equilibrium errors, especially at higher orders of approximation. Finally, we observe that when combined with APEC flux differencing formulations, entropy correction improves robustness for under-resolved solutions and long-time simulations.
The reproducibility repository is available on GitHub and Zenodo.
]]>Although entropy-based summation-by-parts (SBP) discretizations of hyperbolic conservation laws are widely used for their robustness and stability properties, there are very few results on their convergence. We extend a recent convergence analysis of Worku, Del Rey Fernández, and Zingg (2026, DOI: 10.48550/arXiv.2603.18369) in two ways. First, instead of allowing only hyperbolic conservation laws whose fluxes are homogeneous and have globally bounded second derivatives (a restriction essentially to linear or quadratic fluxes), we consider general hyperbolic systems with strictly convex entropy and source terms depending on time and space. Second, instead of requiring a special class of SBP operators, we consider a general framework of diagonal-norm SBP operators on curved meshes, including finite differences, continuous and discontinuous Galerkin methods. Since the error analysis is based on a discrete relative entropy, it is restricted to smooth solutions. To enable a unified treatment of conservation laws, we restrict the analysis to periodic boundary conditions. Numerical results demonstrate that the predicted convergence rates are sharp in general, but can be improved for special cases such as discontinuous Galerkin methods with even polynomial degree and multi-block finite difference methods. An optimal analysis is expected to require more sophisticated arguments specialized to the class of methods instead of the general framework of SBP operators used in this work.
The reproducibility repository is available on GitHub and Zenodo.
]]>We develop structure-preserving numerical methods for the compressible Euler equations, employing potential temperature as a prognostic variable. We construct three numerical fluxes designed to ensure the conservation of entropy and total energy within the discontinuous Galerkin framework on general curvilinear meshes. Furthermore, we introduce a generalization for the kinetic energy preservation property and total energy conservation in the presence of a gravitational potential term. To this end, we adopt a flux-differencing approach for the discretization of the source term, treated as non-conservative product. We present well-balanced schemes for different constant background states for both formulations (total energy and potential temperature) on curvilinear meshes. Finally, we validate the methods by comparing the potential temperature formulation with the traditional Euler equations formulation across a range of classical atmospheric scenarios.
The reproducibility repository is available on GitHub and Zenodo.
As usual, you can find the preprint on arXiv.
]]>The Active Flux method is a numerical method for conservation laws using a combination of cell averages and point values as independent degrees of freedom, based on ideas from finite volumes and finite differences. This unusual mix has been shown to work well in many situations. We expand the theoretical justifications of the Active Flux method by analyzing it from the point of view of summation-by-parts (SBP) operators, which are routinely used to analyze finite difference, finite volume, and finite element schemes. We investigate in what type of setting the Active Flux method can be formulated using classical or degenerate SBP operators, yielding a first and novel approach for showing the energy stability of the Active Flux method. We present the analysis for the one-dimensional scalar linear advection equation with periodic boundary conditions on a uniform grid.
The reproducibility repository is available on GitHub and Zenodo.
As usual, you can find the preprint on arXiv.
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