NDNS+
Day 1 (Tuesday 22 April) @ 13:45–15:15
Julian Köllermeier (University of Groningen)
Beyond Shallow Water Equations: A Hierarchical Approach to Free-Surface Flows via Moment Models
Modeling geophysical free-surface flows is challenging due to the interplay of multiscale dynamics and accuracy requirements. Traditional approaches like the Shallow Water Equations (SWE) are computationally fast but often insufficient for applications demanding greater physical fidelity. Hierarchical moment models offer a promising alternative, deriving an extended PDE system based on a polynomial expansion of the velocity profile. This approach retains the SWE as a base case while enabling enhanced accuracy by incorporating additional equations in a hierarchical way.
In this talk, we present this framework for hierarchical moment models, highlighting their flexibility, structural similarity, and capacity for model adaptivity. We discuss key aspects including (1) the hierarchical nature of the models, (2) the analysis of hyperbolicity and physical properties, and (3) efficient numerical schemes that preserve asymptotic properties. Using numerical examples, we showcase runtime and accuracy improvements in predicting free-surface flows. Additionally, we outline future extensions of the models to account for complex fluids with the aim to bridge the gap between computational efficiency and physical accuracy in free-surface flow simulations.

Karoline Disser (Universität Kassel)
Global solutions and non-trivial long-time behaviour for fluid-elastic interaction
We look at a non-linear system modelling the dynamics of a linearly elastic body immersed in an incompressible viscous fluid in 3d. Assuming no other damping but fluid viscosity, in the case of small initial data, the system admits a unique global strong solution that converges either to a steady state or a “pressure wave”-solution. However, it depends on the geometry of the elastic structure, whether non-trivial pressure waves exist.

Koondanibha Mitra (Eindhoven University of Technology)
Degenerate & singular mixed dimensional diffusion systems: A journey from modelling, and numerical methods, to well-posedness
Nonlinear, degenerate, and coupled (mixed-dimensionally) systems arise in various applications of societal relevance such as biofilm growth, reactive transport in fractured porous media, and cellular biology. For such problems, the biomass or the main-substrate is restricted to a lower dimensional manifold embedded in a domain in, and the evolution of the biomass density/substrate concentration exhibits degenerate and singular diffusion behaviour. The other equations (for nutrients/reagents) are defined on the bulk, and are of linear advection-reaction-diffusion type.
In our analysis, we first propose a backward Euler time-discretisation of the problem where the reactive terms coupling the equations are estimated semi-implicitly, and thus, the equations are dimensionally decoupled and can be solved sequentially. A bulk-surface finite element method is used to discretise the system in space. Then, an iterative linearisation algorithm is proposed to solve the fully-discrete nonlinear problem on discretised. It is proven that the iterations converge unconditionally even for degenerate/singular cases and for curved, thus, showing existence of solution of the fully-discrete nonlinear problem. Then, for flat the convergence of the fully-discrete solutions to the time-discrete solution is shown and order of convergence with respect to mesh-size is estimated. Properties such as well-posedness, boundedness, and positivity of the time-discrete solutions are proven, and the existence of the time-continuous solutions is shown by passing the time-step size to zero. Uniqueness is shown using a contraction argument. Additional properties such as regularity and finite-time blow-up are investigated. Numerical results validate the theoretical predictions. They demonstrate that the iterative solver and the discretization are extremely robust and efficient compared to existing alternatives.
References
- K. Mitra, & S. Sonner (2023). Well-posedness and properties of nonlinear coupled evolution problems modelling biofilm growth. arXiv:2304.00175.
- R.K.H. Smeets, K. Mitra, S. Sonner, & I.S. Pop (2024). Robust time-discretisation and linearisation schemes for singular and degenerate evolution systems modelling biofilm growth. arXiv:2404.00391v2.

Photocredits: Angeline Swinkels