arXiv subjects

Ullika Scholz

Publications and source records attributed to Ullika Scholz.

3 recordsLinked to original sources

arXiv2026

A Splitting Scheme for Dispersive Shallow Moment Equations

The well-known Shallow Water Equations (SWE) are used for modeling incompressible free-surface flows whenever the shallowness allows for a vertical-averaging; i.e., vertical effects are negligible in comparison to horizontal ones. But vertical averaging comes with the price of losing information along the vertical axis. Moment models for shallow flow contain information on the vertical velocity and pressure profile despite being dimensionally reduced. A class of these models incorporating a non-hydrostatic pressure have been introduced before as Dispersive Shallow Moment Models (DSM). However, no method for solving the non-stationary equations has been presented yet, mainly because it was unclear how to compute the pressure equation in the form of the divergence-free constraint. We rewrite the pressure equations of the DSM models in the form of a Poisson-like problem to enable their solution with a projection-type splitting scheme. For the linear equations, we present the calculations for the generalized model and discuss the non-linear case. We state the first two linear models and the corresponding nonlinear counterparts. Finally, we introduce a hybrid Finite-Volume Finite-Difference method and discuss the non-stationary numerical results for an experiment with periodic boundary and uneven bottom topography.

physics.flu-dyn↗

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↗

arXiv2021

Spline Moment Models for the one-dimensional Boltzmann-BGK equation

We introduce Spline Moment Equations (SME) for kinetic equations using a new weighted spline ansatz of the distribution function and investigate the ansatz, the model, and its performance by simulating the one-dimensional Boltzmann-BGK equation. The new basis is composed of weighted constrained splines for the approximation of distribution functions that preserves mass, momentum, and energy. This basis is then used to derive moment equations using a Galerkin approach for a shifted and scaled Boltzmann-BGK equation, to allow for an accurate and efficient discretization in velocity space with an adaptive grid. The equations are given in compact analytical form and we show that the hyperbolicity properties are similar to the well-known Grad moment model. The model is investigated numerically using the shock tube, the symmetric two-beam test and a stationary shock structure test case. All tests reveal the good approximation properties of the new SME model when the parameters of the spline basis functions are chosen properly. The new SME model outperforms existing moment models and results in a smaller error while using a small number of variables for efficient computations.

math.NA↗