Julian Koellermeier

Afroja Parvin, Giovanni Samaey, Julian Koellermeier

We derive the first-moment Shallow Water Exner Moment model with sediment entrainment and deposition (SWEMED1) and show that the full source term has a fully-settled water-at-rest equilibrium manifold. We prove that the model is only weakly hyperbolic at this equilibrium, which prevents the use of Yong's structural stability framework. However, a linear spectral analysis and numerical results do not indicate instability. Based on numerical results, we introduce a fast-slow scaling of the source term, and for the fast limit, we derive a new suspended water-at-rest equilibrium manifold, which has a different structure but is still only weakly hyperbolic. Our results show that the remaining obstruction is linked to the transport closure of the SWEMED1, and we give a constructive direction for the derivation of new closures leading to models with more desirable analytical properties.

Afroja Parvin, Giovanni Samaey, Julian Koellermeier

In this paper, we develop the Hyperbolic Shallow Water Exner Moment model with Erosion and Deposition (HSWEMED), extending the shallow water moment framework to capture coupled morphodynamics with erosion and deposition. HSWEMED introduces a suspended-sediment concentration equation, couples concentration-dependent mixture density with the momentum and higher-order moment equations, and includes source terms due to erosion and deposition. Starting from the incompressible Navier-Stokes equations for a water-sediment mixture, we derive a coupled system consisting of the shallow water equations, moment equations for polynomial velocity coefficients, a depth-averaged suspended-sediment equation, and an Exner equation for bedload transport with erosion-deposition coupling. Although the transported scalar is depth-averaged, we reconstruct a low-order vertical concentration profile consistent with the moment representation of velocity, providing the near-bed concentration needed in the closure. We prove hyperbolicity through hyperbolic regularization and derive dissipative energy balance relations for lower-order models. Numerical results are obtained with a path-conservative finite-volume scheme based on a Lax-Friedrichs-type flux. Several dam-break tests, including wet/dry front cases, are validated against laboratory experiments, showing improved accuracy over existing shallow water moment models. The proposed HSWEMED provides a mathematically well-posed and computationally efficient framework for morphodynamic simulations.

Julian Koellermeier

Shallow Water Moment Equations are reduced-order models for free-surface flows that employ a vertical velocity expansion and derive additional so-called moment equations for the expansion coefficients. Among desirable analytical properties for such systems of equations are hyperbolicity, accuracy, correct momentum equation, and interpretable steady states. In this paper, we show analytically that existing models fail at different of these properties and we derive new models overcoming the disadvantages. This is made possible by performing a hyperbolic regularization not in the convective variables (as done in the existing models) but in the primitive variables. Via analytical transformations between the convective and primitive system, we can prove hyperbolicity and compute analytical steady states of the new models. Simulating a dam-break test case, we demonstrate the accuracy of the new models and show that it is essential for accuracy to preserve the momentum equation.

Antoine Oriou, Philipp Krah, Julian Koellermeier

This paper introduces the Intrinsic Dimension Estimating Autoencoder (IDEA), which identifies the underlying intrinsic dimension of a wide range of datasets whose samples lie on either linear or nonlinear manifolds. Beyond estimating the intrinsic dimension, IDEA is also able to reconstruct the original dataset after projecting it onto the corresponding latent space, which is structured using re-weighted double CancelOut layers. Our key contribution is the introduction of the projected reconstruction loss term, guiding the training of the model by continuously assessing the reconstruction quality under the removal of an additional latent dimension. We first assess the performance of IDEA on a series of theoretical benchmarks to validate its robustness. These experiments allow us to test its reconstruction ability and compare its performance with state-of-the-art intrinsic dimension estimators. The benchmarks show good accuracy and high versatility of our approach. Subsequently, we apply our model to data generated from the numerical solution of a vertically resolved one-dimensional free-surface flow, following a pointwise discretization of the vertical velocity profile in the horizontal direction, vertical direction, and time. IDEA succeeds in estimating the dataset's intrinsic dimension and then reconstructs the original solution by working directly within the projection space identified by the network.