arXiv subjects

Juliane Rosemeier

Publications and source records attributed to Juliane Rosemeier.

4 recordsLinked to original sources

arXiv2026

A WKB-related time-stepping scheme for differential equations describing oscillatory systems

In this study, we present a novel time-stepping scheme for multiscale differential equations describing oscillatory systems with well-separated scales, where the scale separation is controlled by a small parameter $\epsilon$. The time-stepping method is related to a multi-modal WKB approximation and relies on a transformation of variables derived in this work. The analysis reveals that, in the transformed formulation, the leading-order oscillations are either eliminated or appear only at higher asymptotic order. The method is applied to ordinary differential equations, including the well-known van der Pol oscillator. We investigate the accuracy of the proposed method numerically for different parameter regimes, in particular for decreasing values of $\epsilon$, and study how the parameters of the numerical scheme must be adapted as $\epsilon$ is reduced. In the presented numerical tests, the computational cost remains bounded as $\epsilon$ is decreased.

math.NA↗

arXiv2023

Multi-level Parareal algorithm with Averaging for Oscillatory Problems

The present study is an extension of the work done in Parareal convergence for oscillatory pdes with finite time-scale separation (2019), A. G. Peddle, T. Haut, and B. Wingate, [16], and An asymptotic parallel-in-time method for highly oscillatory pdes (2014), T. Haut and B. Wingate, [10], where a two-level Parareal method with averaging is examined. The method proposed in this paper is a multi-level Parareal method with arbitrarily many levels, which is not restricted to the two-level case. We give an asymptotic error estimate which reduces to the two-level estimate for the case when only two levels are considered. Introducing more than two levels has important consequences for the averaging procedure, as we choose separate averaging windows for each of the different levels, which is an additional new feature of the present study. The different averaging windows make the proposed method especially appropriate for multi-scale problems, because we can introduce a level for each intrinsic scale of the problem and adapt the averaging procedure such that we reproduce the behavior of the model on the particular scale resolved by the level. The computational complexity of the new method is investigated and the efficiency is studied on several examples.

math.NA↗

arXiv2021

Pattern formation in clouds via Turing instabilities

Pattern formation in clouds is a well-known feature, which can be observed almost every day. However, the guiding processes for structure formation are mostly unknown, and also theoretical investigations of cloud patterns are quite rare. From many scientific disciplines the occurrence of patterns in non-equilibrium systems due to Turing instabilities is known, i.e. unstable modes grow and form spatial structures. In this study we investigate a generic cloud model for the possibility of Turing instabilities. For this purpose, the model is extended by diffusion terms. We can show that for some cloud models, i.e special cases of the generic model, no Turing instabilities are possible. However, we also present a general class of cloud models, where Turing instabilities can occur. A key requisite is the occurrence of (weakly) nonlinear terms for accretion. Using numerical simulations for a special case of the general class of cloud models, we show spatial patterns of clouds in one and two spatial dimensions. From the numerical simulations we can see that the competition between collision terms and sedimentation is an important issue for the existence of pattern formation.

math.DS↗

arXiv2018

Intercomparison of Warm-Rain Bulk Microphysics Schemes using Asymptotics

Clouds are important components of the atmosphere. Since it is usually not possible to treat them as ensembles of huge numbers of particles, parameterizations on the basis of averaged quantities (mass and/or number concentration) must be derived. Since no first-principles derivations of such averaged schemes are available today, many alternative approximating schemes of cloud processes exist. Most of these come in the form of nonlinear differential equations. It is unclear whether these different cloud schemes behave similarly under controlled local conditions, and much less so when they are embedded dynamically in a full atmospheric flow model. We use mathematical methods from the theory of dynamical systems and asymptotic analysis to compare two operational cloud schemes and one research scheme qualitatively in a simplified context in which the moist dynamics is reduced to a system of ODEs. It turns out that these schemes behave qualitatively differently on shorter time scales, whereas at least their long time behavior is similar under certain conditions. These results show that the quality of computational forecasts of moist atmospheric flows will generally depend strongly on the formulation of the cloud schemes used.

physics.ao-ph↗