08 Fakultät Mathematik und Physik
Permanent URI for this collectionhttps://elib.uni-stuttgart.de/handle/11682/9
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Item Open Access On a stochastic Camassa-Holm type equation with higher order nonlinearities(2020) Rohde, Christian; Tang, HaoThe subject of this paper is a generalized Camassa-Holm equation under random perturbation. We first establish local existence and uniqueness results as well as blow-up criteria for pathwise solutions in the Sobolev spaces Hs with s>3/2. Then we analyze how noise affects the dependence of solutions on initial data. Even though the noise has some already known regularization effects, much less is known concerning the dependence on initial data. As a new concept we introduce the notion of stability of exiting times and construct an example showing that multiplicative noise (in Itô sense) cannot improve the stability of the exiting time, and simultaneously improve the continuity of the dependence on initial data. Finally, we obtain global existence theorems and estimate associated probabilities.Item Open Access On the stochastic Dullin-Gottwald-Holm equation : global existence and wave-breaking phenomena(2020) Rohde, Christian; Tang, HaoWe consider a class of stochastic evolution equations that include in particular the stochastic Camassa-Holm equation. For the initial value problem on a torus, we first establish the local existence and uniqueness of pathwise solutions in the Sobolev spaces Hs with s>3/2. Then we show that strong enough nonlinear noise can prevent blow-up almost surely. To analyze the effects of weaker noise, we consider a linearly multiplicative noise with non-autonomous pre-factor. Then, we formulate precise conditions on the initial data that lead to global existence of strong solutions or to blow-up. The blow-up occurs as wave breaking. For blow-up with positive probability, we derive lower bounds for these probabilities. Finally, the blow-up rate of these solutions is precisely analyzed.Item Open Access A posteriori error analysis and adaptive non-intrusive numerical schemes for systems of random conservation laws(2020) Giesselmann, Jan; Meyer, Fabian; Rohde, ChristianThis article considers one-dimensional random systems of hyperbolic conservation laws. Existence and uniqueness of random entropy admissible solutions for initial value problems of conservation laws, which involve random initial data and random flux functions, are established. Based on these results an a posteriori error analysis for a numerical approximation of the random entropy solution is presented. For the stochastic discretization, a non-intrusive approach, namely the Stochastic Collocation method is used. The spatio-temporal discretization relies on the Runge-Kutta Discontinuous Galerkin method. The a posteriori estimator is derived using smooth reconstructions of the discrete solution. Combined with the relative entropy stability framework this yields computable error bounds for the entire space-stochastic discretization error. The estimator admits a splitting into a stochastic and a deterministic (space-time) part, allowing for a novel residual-based space-stochastic adaptive mesh refinement algorithm. The scaling properties of the residuals are investigated and the efficiency of the proposed adaptive algorithms is illustrated in various numerical examples.Item Open Access Towards hybrid two‐phase modelling using linear domain decomposition(2022) Seus, David; Radu, Florin A.; Rohde, ChristianThe viscous flow of two immiscible fluids in a porous medium on the Darcy scale is governed by a system of nonlinear parabolic equations. If infinite mobility of one phase can be assumed (e.g., in soil layers in contact with the atmosphere) the system can be substituted by the scalar Richards model. Thus, the porous medium domain may be partitioned into disjoint subdomains where either the full two‐phase or the simplified Richards model dynamics are valid. Extending the previously considered one‐model situations we suggest coupling conditions for this hybrid model approach. Based on an Euler implicit discretization, a linear iterative (L‐type) domain decomposition scheme is proposed, and proved to be convergent. The theoretical findings are verified by a comparative numerical study that in particular confirms the efficiency of the hybrid ansatz as compared to full two‐phase model computations.Item Open Access Compressible multicomponent flow in porous media with Maxwell‐Stefan diffusion(2020) Ostrowski, Lukas; Rohde, ChristianWe introduce a Darcy‐scale model to describe compressible multicomponent flow in a fully saturated porous medium. In order to capture cross‐diffusive effects between the different species correctly, we make use of the Maxwell–Stefan theory in a thermodynamically consistent way. For inviscid flow, the model turns out to be a nonlinear system of hyperbolic balance laws. We show that the dissipative structure of the Maxwell‐Stefan operator permits to guarantee the existence of global classical solutions for initial data close to equilibria. Furthermore, it is proven by relative entropy techniques that solutions of the Darcy‐scale model tend in a certain long‐time regime to solutions of a parabolic limit system.Item Open Access Error control for statistical solutions of hyperbolic systems of conservation laws(2021) Giesselmann, Jan; Meyer, Fabian; Rohde, ChristianStatistical solutions have recently been introduced as an alternative solution framework for hyperbolic systems of conservation laws. In this work, we derive a novel a posteriori error estimate in the Wasserstein distance between dissipative statistical solutions and numerical approximations obtained from the Runge-Kutta Discontinuous Galerkin method in one spatial dimension, which rely on so-called regularized empirical measures. The error estimator can be split into deterministic parts which correspond to spatio-temporal approximation errors and a stochastic part which reflects the stochastic error. We provide numerical experiments which examine the scaling properties of the residuals and verify their splitting.Item Open Access Reduced-dimensional modelling for nonlinear convection-dominated flow in cylindric domains(2024) Mel’nyk, Taras; Rohde, ChristianThe aim of the paper is to construct and justify asymptotic approximations for solutions to quasilinear convection-diffusion problems with a predominance of nonlinear convective flow in a thin cylinder, where an inhomogeneous nonlinear Robin-type boundary condition involving convective and diffusive fluxes is imposed on the lateral surface. The limit problem for vanishing diffusion and the cylinder shrinking to an interval is a nonlinear first-order conservation law. For a time span that allows for a classical solution of this limit problem corresponding uniform pointwise and energy estimates are proven. They provide precise model error estimates with respect to the small parameter that controls the double viscosity-geometric limit. In addition, other problems with more higher Péclet numbers are also considered.Item Open Access Investigation of crystal growth in enzymatically induced calcite precipitation by micro-fluidic experimental methods and comparison with mathematical modeling(2021) Wolff, Lars von; Weinhardt, Felix; Class, Holger; Hommel, Johannes; Rohde, ChristianEnzymatically induced calcite precipitation (EICP) is an engineering technology that allows for targeted reduction of porosity in a porous medium by precipitation of calcium carbonates. This might be employed for reducing permeability in order to seal flow paths or for soil stabilization. This study investigates the growth of calcium-carbonate crystals in a micro-fluidic EICP setup and relies on experimental results of precipitation observed over time and under flow-through conditions in a setup of four pore bodies connected by pore throats. A phase-field approach to model the growth of crystal aggregates is presented, and the corresponding simulation results are compared to the available experimental observations. We discuss the model’s capability to reproduce the direction and volume of crystal growth. The mechanisms that dominate crystal growth are complex depending on the local flow field as well as on concentrations of solutes. We have good agreement between experimental data and model results. In particular, we observe that crystal aggregates prefer to grow in upstream flow direction and toward the center of the flow channels, where the volume growth rate is also higher due to better supply.Item Open Access Mathematical challenges for the theory of hyperbolic balance laws in fluid mechanics : complexity, scales, randomness(2024) Lukáčová-Medvid’ová, Mária; Rohde, ChristianUnderstanding the dynamics of hyperbolic balance laws is of paramount interest in the realm of fluid mechanics. Nevertheless, fundamental questions on the analysis and the numerics for distinctive hyperbolic features related to turbulent flow motion remain vastly open. Recent progress on the mathematical side reveals novel routes to face these concerns. This includes findings about the failure of the entropy principle to ensure uniqueness, the use of structure-preserving concepts in high-order numerical methods, and the advent of tailored probabilistic approaches. Whereas each of these three directions on hyperbolic modelling are of completely different origin they are all linked to small- or subscale features in the solutions which are either enhanced or depleted by the hyperbolic nonlinearity. Thus, any progress in the field might contribute to a deeper understanding of turbulent flow motion on the basis of the continuum-scale mathematical models. We present an overview on the mathematical state-of-the-art in the field and relate it to the scientific work in the DFG Priority Research Programme 2410. As such, the survey is not necessarily targeting at readers with comprehensive knowledge on hyperbolic balance laws but instead aims at a general audience of reseachers which are interested to gain an overview on the field and associated challenges in fluid mechanics.