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C. Klingenberg

Publications and source records attributed to C. Klingenberg.

9 recordsLinked to original sources
arXiv2022

A finite-volume scheme for modeling compressible magnetohydrodynamic flows at low Mach numbers in stellar interiors

Fully compressible magnetohydrodynamic (MHD) simulations are a fundamental tool for investigating the role of dynamo amplification in the generation of magnetic fields in deep convective layers of stars. The flows that arise in such environments are characterized by low (sonic) Mach numbers (M_son < 0.01 ). In these regimes, conventional MHD codes typically show excessive dissipation and tend to be inefficient as the Courant-Friedrichs-Lewy (CFL) constraint on the time step becomes too strict. In this work we present a new method for efficiently simulating MHD flows at low Mach numbers in a space-dependent gravitational potential while still retaining all effects of compressibility. The proposed scheme is implemented in the finite-volume Seven-League Hydro (SLH) code, and it makes use of a low-Mach version of the five-wave Harten-Lax-van Leer discontinuities (HLLD) solver to reduce numerical dissipation, an implicit-explicit time discretization technique based on Strang splitting to overcome the overly strict CFL constraint, and a well-balancing method that dramatically reduces the magnitude of spatial discretization errors in strongly stratified setups. The solenoidal constraint on the magnetic field is enforced by using a constrained transport method on a staggered grid. We carry out five verification tests, including the simulation of a small-scale dynamo in a star-like environment at M_son ~ 0.001 . We demonstrate that the proposed scheme can be used to accurately simulate compressible MHD flows in regimes of low Mach numbers and strongly stratified setups even with moderately coarse grids.

astro-ph.SR↗
arXiv2021

Well-balanced treatment of gravity in astrophysical fluid dynamics simulations at low Mach numbers

Accurate simulations of flows in stellar interiors are crucial to improving our understanding of stellar structure and evolution. Because the typically slow flows are merely tiny perturbations on top of a close balance between gravity and the pressure gradient, such simulations place heavy demands on numerical hydrodynamics schemes. We demonstrate how discretization errors on grids of reasonable size can lead to spurious flows orders of magnitude faster than the physical flow. Well-balanced numerical schemes can deal with this problem. Three such schemes were applied in the implicit, finite-volume Seven-League Hydro (SLH) code in combination with a low-Mach-number numerical flux function. We compare how the schemes perform in four numerical experiments addressing some of the challenges imposed by typical problems in stellar hydrodynamics. We find that the $\alpha$-$\beta$ and deviation well-balancing methods can accurately maintain hydrostatic solutions provided that gravitational potential energy is included in the total energy balance. They accurately conserve minuscule entropy fluctuations advected in an isentropic stratification, which enables the methods to reproduce the expected scaling of convective flow speed with the heating rate. The deviation method also substantially increases accuracy of maintaining stationary orbital motions in a Keplerian disk on long timescales. The Cargo-LeRoux method fares substantially worse in our tests, although its simplicity may still offer some merits in certain situations. Overall, we find the well-balanced treatment of gravity in combination with low Mach number flux functions essential to reproducing correct physical solutions to challenging stellar slow-flow problems on affordable collocated grids.

astro-ph.SR↗
arXiv2020

Using the Navier-Cauchy equation for motion estimation in dynamic imaging

Tomographic image reconstruction is well understood if the specimen being studied is stationary during data acquisition. However, if this specimen changes during the measuring process, standard reconstruction techniques can lead to severe motion artefacts in the computed images. Solving a dynamic reconstruction problem therefore requires to model and incorporate suitable information on the dynamics in the reconstruction step to compensate for the motion. Many dynamic processes can be described by partial differential equations which thus could serve as additional information for the purpose of motion compensation. In this article, we consider the Navier-Cauchy equation which characterizes small elastic deformations and serves, for instance, as a model for respiratory motion. Our goal is to provide a proof-of-concept that by incorporating the deformation fields provided by this PDE, one can reduce the respective motion artefacts in the reconstructed image. To this end, we solve the Navier-Cauchy equation prior to the image reconstruction step using suitable initial and boundary data. Then, the thus computed deformation fields are incorporated into an analytic dynamic reconstruction method to compute an image of the unknown interior structure. The feasibility is illustrated with numerical examples from computerized tomography.

math.AP↗
arXiv2018

Arbitrary order finite volume well-balanced schemes for the Euler equations with gravity

This work presents arbitrary high order well balanced finite volume schemes for the Euler equations with a prescribed gravitational field. It is assumed that the desired equilibrium solution is known, and we construct a scheme which is exactly well balanced for that particular equilibrium. The scheme is based on high order reconstructions of the fluctuations from equilibrium of density, momentum and pressure, and on a well balanced integration of the source terms, while no assumptions are needed on the numerical flux, beside consistency. This technique allows to construct well balanced methods also for a class of moving equilibria. Several numerical tests demonstrate the performance of the scheme on different scenarios, from equilibrium solutions to non steady problems involving shocks. The numerical tests are carried out with methods up to fifth order in one dimension, and third order accuracy in 2D.

math.NA↗
arXiv2018

Kinetic/Fluid micro-macro numerical scheme for a two component gas mixture

This work is devoted to the numerical simulation of the \BGK equation for two species in the fluid limit using a particle method. Thus, we are interested in a gas mixture consisting of two species without chemical reactions assuming that the number of particles of each species remains constant. We consider the kinetic two species model proposed by Klingenberg, Pirner and Puppo in 2017, which separates the intra and interspecies collisions. We want to study numerically the influence of the two relaxation term, one corresponding to intra, the other to interspecies collisions. For this, we use the method of micro-macro decomposition. First, we derive an equivalent model based on the micro-macro decomposition (see Bennoune, Lemou and Mieussens, 2007 and Crestetto, Crouseilles and Lemou, 2013). The kinetic micro part is solved by a particle method, whereas the fluid macro part is discretized by a standard finite volume scheme. Main advantages of this approach are: (i) the noise inherent to the particle method is reduced compared to a standard (without micro-macro decomposition) particle method, (ii) the computational cost of the method is reduced in the fluid limit since a small number of particles is then sufficient.

math.AP↗
arXiv2018

On quantifying uncertainties for the linearized BGK kinetic equation

We consider the linearized BGK equation and want to quantify uncertainties in the case of modelling errors. More specifically, we want to quantify the error produced if the pre-determined equilibrium function is chosen inaccurately. In this paper we consider perturbations in the velocity and in the temperature of the equilibrium function and consider how much the error is amplified in the solution.

math.AP↗
arXiv2018

Existence, Uniqueness and Positivity of solutions for BGK models for mixtures

We consider kinetic models for a multi component gas mixture without chemical reactions. In the literature, one can find two types of BGK models in order to describe gas mixtures. One type has a sum of BGK type interaction terms in the relaxation operator, for example the model described by Klingenberg, Pirner and Puppo, 2017 which contains well-known models of physicists and engineers for example Hamel, 1965, and Gross and Krook, 1956, as special cases. The other type contains only one collision term on the right-hand side, for example the well-known model of Andries, Aoki and Perthame, 2002. For each of these two models we prove existence, uniqueness and positivity of solutions in the first part of the paper. In the second part, we use the first model in order to determine an unknown function in the energy exchange of the macroscopic equations for gas mixtures described by Dellacherie.

math.AP↗
arXiv2009

Thermonuclear explosions of rapidly rotating white dwarfs - I. Deflagrations

Context: Turbulent deflagrations of Chandrasekhar mass White Dwarfs are commonly used to model Type Ia Supernova explosions. In this context, rapid rotation of the progenitor star is plausible but has so far been neglected. Aims: The aim of this work is to explore the influence of rapid rotation on the deflagration scenario. Methods: We use three dimensional hydrodynamical simulations to model turbulent deflagrations ignited within a variety of rapidly rotating CO WDs obeying rotation laws suggested by accretion studies. Results: We find that rotation has a significant impact on the explosion. The flame develops a strong anisotropy with a preferred direction towards the stellar poles, leaving great amounts of unburnt matter along the equatorial plane. Conclusions: The large amount of unburnt matter is contrary to observed spectral features of SNe Ia. Thus, rapid rotation of the progenitor star and the deflagration scenario are incompatible in order to explain SNe Ia.

astro-ph.HE↗