arXiv subjects

Julian Koellermeier

Publications and source records attributed to Julian Koellermeier.

At least 19 recordsLinked to original sources

arXiv2026

On Hyperbolic Stochastic Galerkin Projections of Shallow Water Linearised Moment Equations

In this work, we present an intrusive stochastic Galerkin formulation of the one-dimensional shallow water linearised moment equations expressed in conservative variables, using the pseudospectral product for generalised polynomial chaos expansions. The shallow water linearised moment equations constitute a hyperbolic system of partial differential equations with an arbitrary number of equations that enhance the accuracy of the standard shallow water equations. Without loss of generality, we assume for both the theoretical analysis and the simulations that the uncertain parameter is the friction coefficient. For the new stochastic Galerkin shallow water linearised moment equations, we derive an energy equation, analyse the hyperbolicity - since this property is not preserved by the stochastic Galerkin projection - and introduce a regularisation to ensure hyperbolicity for the linear case of the shallow water linearised moment equations. Through numerical tests, we demonstrate the accuracy of the new stochastic Galerkin formulation in comparison with a non-intrusive Monte Carlo method, showing that the stochastic Galerkin approach achieves comparable accuracy with significantly faster run times.

math.NA↗

arXiv2026

Entropy analysis and entropy stable DG methods for the 1D shallow water moment equations

We demonstrate that the one-dimensional shallow water moment equations satisfy an auxiliary entropy conservation law, where the entropy function corresponds to the total energy. Additionally, we show that the classical Newtonian slip friction and Manning friction terms are entropy dissipative with respect to the developed entropy variables. The results from the continuous entropy analysis are used to construct an entropy stable and well-balanced nodal discontinuous Galerkin spectral element method for the spatial approximation. Key to ensure the entropy stability of the scheme is the derivation of entropy conservative numerical fluxes that satisfy a discrete version of the entropy flux compatibility condition. Finally, numerical examples demonstrate the performance of the scheme and validate the theoretical results.

math.NA↗

arXiv2026

Water-at-Rest Equilibrium Stability Analysis of a first-moment Shallow Water Exner Moment Model with Sediment Entrainment and Deposition: Extended Technical Report

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.

math.AP↗

arXiv2026

A Hyperbolic Moment Based Shallow Water Model for Coupled Bedload Suspended Load Morphodynamics with Variable Density

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.

math.NA↗

arXiv2026

Primitive variable regularization to derive novel Hyperbolic Shallow Water Moment Equations

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.

math.NA↗

arXiv2026

Asymptotic Analysis of Shallow Water Moment Equations

The Shallow Water Moment Equations (SWME) are an extension of the Shallow Water Equations (SWE) for improved modelling of free-surface flows. In contrast to the SWE, the SWME incorporate vertical velocity profile information. The SWME framework approximates vertical velocity profiles using a polynomial expansion with Legendre polynomials and polynomial coefficients, also called moment variables. The SWME have an increased number of variables that must always be incorporated, even when the flow approaches a viscous slip equilibrium state that could be characterised by vanishing moment variables. To reduce the complexity of the SWME in cases proximate to this equilibrium, we conduct an asymptotic analysis of the SWME. This yields the closed form Reduced Shallow Water Moment Equations (RSWME) for deviations from the equilibrium. The RSWME have fewer variables, compared to the SWME. The hyperbolicity of the RSWME is analysed. Numerical tests include a wave with a sharp height gradient, a smoother height gradient and a square root velocity profile. The numerical tests demonstrate that the RSWME reduce computational cost up to 77% compared to the SWME and improves accuracy up to 88% over the SWE.

math.NA↗

arXiv2026

A new Energy Equation Derivation for the Shallow Water Linearized Moment Equations

Shallow Water Moment Equations (SWME) are extensions to the well-known Shallow Water Equations (SWE) for the efficient modeling and numerical simulation of free-surface flows. While the SWE typically assume a depth-averaged vertical velocity profile, the SWME allow for vertical variations of the velocity profile. The SWME therefore assume a polynomial profile and then derive additional evolution equations for the polynomial coefficients via higher order depth integration. In this work, we perform a new systematic derivation of the energy equation for a specific variant of the SWME, called the Shallow Water Linearized Moment Equations (SWLME). The derivation is based on the standard SWE energy equation derivation and includes the skew-symmetric formulation of the model. The new systematic derivation is beneficial for the extension to other SWME variants and their numerical solution.

math.NA↗

arXiv2026

Intrinsic Dimension Estimating Autoencoder (IDEA) Using CancelOut Layer and a Projected Loss

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.

cs.LG↗

arXiv2026

Explicit complex time integrators for stiff problems

Most numerical methods for time integration use real-valued time steps. Complex time steps, however, can provide an additional degree of freedom, as we can select the magnitude of the time step in both the real and imaginary directions. We show that specific paths in the complex time plane lead to expanded stability regions, providing clear computational advantages for complex-valued systems. In particular, we highlight the Schrödinger equation, for which complex time integrators can be uniquely optimal. Furthermore, we demonstrate that these benefits extend to certain classes of real-valued stiff systems by coupling complex time steps with the Projective Integration method.

math.NA↗

arXiv2025

Analysis of kinetic-diffusion Monte Carlo simulation and source term estimation scheme in nuclear fusion applications

In plasma edge simulations, the behavior of neutral particles is often described by a Boltzmann--BGK equation. Solving this kinetic equation and estimating the moments of its solution are essential tasks, typically carried out using Monte Carlo (MC) methods. However, for large-sized reactors, like ITER and DEMO, high collision rates lead to a substantial computational cost. To accelerate the calculation, an asymptotic-preserving kinetic-diffusion Monte Carlo (KDMC) simulation method (Mortier et al., SIAM J. Sci. Comput., 2022) and a corresponding fluid estimation technique (Mortier et al., Contrib. Plasma Phys., 2022) have recently been proposed. In this work, we present a comprehensive analysis of the convergence of KDMC combined with the associated fluid estimation. The analysis consists of proving theoretical upper bounds for both KDMC and the fluid estimation, and numerical verifications of these bounds. In addition, we compare the analyzed algorithm with a purely fluid-based method using the fully kinetic MC method as a reference. The algorithm consistently achieves lower error than the fluid-based method, and even one order of magnitude lower in a fusion-relevant test case. Moreover, the algorithm exhibits a significant speedup compared to the reference kinetic MC method. Overall, our analysis confirms the effectiveness of KDMC with the associated fluid estimation in nuclear fusion applications.

math.NA↗

arXiv2025

A moment model of shallow granular flows with variable friction laws

In this work, we develop a modelling framework for granular flows based on the shallow water moment equations on inclined planes. Under the assumption of a polynomial expansion of the velocity field, the model extends the classical shallow water equations to vertically variable velocity profiles. The friction effects, which are captured through the strain-rate tensor, are incorporated into the model in two terms, the bulk and bottom friction. We propose a modelling procedure to incorporate general friction laws into our framework and exemplify this combining the Manning, Coulomb, Savage-Hutter, and $μ(I)$-rheology friction models in our modeling framework. Moreover, we develop a path-conservative finite volume numerical scheme based on the polynomial viscosity matrix method to properly handle the stiffness of the source terms. Numerical simulations are presented for different models of friction, including the case of wet-dry fronts.

math.NA↗

arXiv2025

Moment Approximations to Magnetic Rotating Shallow Flows

Originally introduced to describe a transition region in stars, the magnetic rotating shallow water (MRSW) model is now used in many solar physics and geophysical applications. Derived from the 3-D incompressible magnetohydrodynamic system, the shallow nature of these applications motivates depth-averaging of both the velocities and magnetic fields. This is advantageous in terms of computational efficiency -- but at the loss of vertical information, thus limiting the predictive power of the MRSW model. To overcome this problem, we employ higher-order vertical moments, but now in the context of conductive fluids. In doing so, the new approximation maintains non-constant vertical profiles of both the horizontal magnetic fields and horizontal velocities, while still remaining in the simplified 2-D framework corresponding to depth integration. In this work, we extend the derivation of the shallow water moment equations to derive the MRSW moment system of arbitrary order; i.e., we represent the vertical profiles of the velocities -- and now additionally the magnetic fields -- by arbitrary-order polynomial expansions, and close the new expanded 2-D system with evolution equations for these polynomial coefficients, found via Galerkin projection. Through numerical experiments for MRSW moment systems up to third-order, we demonstrate that these moment approximations reduce model error without significantly sacrificing computational efficiency.

math.NA↗

arXiv2025

Model-Adaptive Simulation of Hierarchical Shallow Water Moment Equations in One Dimension

Shallow free surface flows are often characterized by both subdomains that require high modeling complexity and subdomains that can be sufficiently accurately modeled with low modeling complexity. Moreover, these subdomains may change in time as the water flows through the domain. This motivates the need for space and time adaptivity in the simulation of shallow free surface flows. In this paper, we develop the first adaptive simulations using the recently developed Shallow Water Moment Equations, which are an extension of the standard Shallow Water Equations that allow for vertically changing velocity profiles by including additional variables and equations. The model-specific modeling complexity of a shallow water moment model is determined by its order. The higher the order of the model, the more variables and equations are included in the model. Shallow water moment models are ideally suited for adaptivity because they are hierarchical such that low-order models and high-order models share the same structure. To enable adaptive simulations, we propose two approaches for the coupling of the varying-order shallow water moment equations at their boundary interfaces. The first approach dynamically updates padded state variables but cannot be written in conservative form, while the second approach uses fixed padded state variable values of zero and reduces to conservative form for conservative moment equations. The switching procedure between high-order models and low-order models is based on a new set of model error estimators, originating from a decomposition of the high-order models. Numerical results of the collision of a dam-break wave with a smooth wave yield accurate results, while achieving speedups up to 60 percent compared to a non-adaptive model with fixed modeling complexity.

physics.flu-dyn↗

arXiv2025

Numerical analysis of fluid estimation for source terms in neutral particles simulation

In plasma edge simulations, kinetic Monte Carlo (MC) is often used to simulate neutral particles and estimate source terms. For large-sized reactors, like ITER and DEMO, high particle collision rates lead to a substantial computational cost for such schemes. To address this challenge, an asymptotic-preserving kinetic-diffusion Monte Carlo (KDMC) simulation method and a corresponding fluid estimation technique have been proposed in the literature. In this work, we perform numerical analysis on the convergence of KDMC with the fluid estimation. To do so, we compare the accuracy of the analyzed algorithm with the accuracy of an approximate fluid method using the kinetic MC method as a reference. In a one-dimensional test case, KDMC with the fluid estimation achieves at least one order of magnitude lower errors than the fluid method for both high- and low-collisional regimes. Moreover, KDMC with the fluid estimation outperforms the kinetic MC method with a clear speed-up. Overall, our analysis confirms the effectiveness of the discussed algorithm.

cs.CE↗

arXiv2025

Spline Shallow Water Moment Equations

Reduced models for free-surface flows are required due to the high dimensionality of the underlying incompressible Navier-Stokes equations, which need to fully resolve the flow in vertical direction to compute the surface height. On the other hand, standard reduced models, such as the classical Shallow Water Equations (SWE), which assume a small depth-to-length ratio and use depth-averaging, do not provide information about the vertical velocity profile variations. As a compromise, a recently proposed moment approach for shallow flow using Legendre polynomials as ansatz functions for vertical velocity variations showed the derivation of so-called Shallow Water Moment Equations (SWME) that combine low dimensionality with velocity profile modeling. However, only global polynomials are considered so far. This paper introduces Spline Shallow Water Moment Equations (SSWME) where piecewise defined spline ansatz functions allow for a flexible representation of velocity profiles with lower regularity. The local support of the spline basis functions opens up the possibility of adaptability and greater flexibility regarding some typical profile shapes. We systematically derive and analyze hierarchies of SSWME models with different number of basis functions and different degrees, before deriving a regularized hyperbolic version by performing a hyperbolic regularization with analytical proof of hyperbolicity for a hierarchy of high-order SSWME models. Numerical simulations show high accuracy and robustness of the new models.

math.NA↗

arXiv2025

Well-balanced path-conservative discontinuous Galerkin methods with equilibrium preserving space for shallow water linearized moment equations

This paper presents high-order, well-balanced, path-conservative discontinuous Galerkin (DG) methods for the shallow water linearized moment equations (SWLME), designed to preserve both still and moving water equilibrium states. Unlike the multi-layer shallow water equations, which model vertical velocity variations using multiple distinct layers, the SWLME employs a polynomial expansion of velocity profiles with up to $N$ moments. This approach enables a more detailed representation of vertical momentum transfer and complex velocity profiles while retaining hyperbolicity. However, the presence of non-conservative terms and complex steady-state structures introduces significant numerical challenges. Addressing these challenges, we develop path-conservative DG schemes grounded in the Dal Maso-LeFloch-Murat (DLM) theory for non-conservative products. Our method balances flux gradients, non-conservative terms, and source terms through equilibrium-preserving spaces. For the still water equilibrium, we reformulate the equations into a quasilinear form that eliminates source terms, inherently preserving steady states. For the moving water equilibrium, we extend the DG method by transforming conservative variables into equilibrium variables and employing linear segment paths. Theoretical analysis and numerical experiments demonstrate that the proposed methods achieve exact equilibrium preservation while maintaining high-order accuracy, even in scenarios with vertical velocity variations and complex topographies.

math.NA↗

arXiv2025

High order global flux schemes for general steady state preservation of shallow water moment equations with non-conservative products

Shallow water moment equations are reduced-order models for free-surface flows that allow to represent vertical variations of the velocity profile at the expense of additional evolution equations for a number of additional variables, so called moments. This introduces non-linear non-conservative products in the system, which make the analytical characterization of steady states much harder if not impossible. The lack of analytical steady states poses a challenge for the design of well-balanced schemes, which aim at preserving such steady states as crucial in many applications. In this work, we present a family of fully well-balanced, high-order WENO finite volume methods for general hyperbolic balance laws with non-conservative products like the shallow water moment equations, for which no analytical steady states are available. The schemes are based on the flux globalization approach, in which both source terms and non-conservative products are integrated with a tailored high order quadrature in the divergence term. The resulting global flux is then reconstructed instead of the conservative variables to preserve all steady states. Numerical tests show the optimal convergence of the method and a significant error reduction for steady state solutions. Furthermore, we provide a numerical comparison of perturbed steady states for different families of shallow water moment equations, which illustrates the flexibility of our method that is valid for general equations without prior knowledge of steady states.

math.NA↗

arXiv2025

Flux Globalization Based Well-Balanced Path-Conservative Central-Upwind Schemes for Shallow Water Linearized Moment Equations

We develop second-order path-conservative central-upwind (PCCU) schemes for the hyperbolic shallow water linearized moment equations (HSWLME), which are an extension of standard depth-averaged models for free-surface flows. The proposed PCCU schemes are constructed via flux globalization strategies adapted to the nonconservative form via a path-conservative finite-volume method. The resulting scheme is well-balanced (WB) in the sense that it is capable of exactly preserving physically relevant steady states including moving-water ones. We validate the proposed scheme on several benchmarks, including smooth solutions, small perturbation of steady states, and dam-break scenarios. These results demonstrate that our flux globalization based WB PCCU schemes provide a reliable framework for computing solutions of shallow water moment models with nonlinear and nonconservative features.

math.NA↗