arXiv subjects

Siegfried Müller

Publications and source records attributed to Siegfried Müller.

17 recordsLinked to original sources

arXiv2026

Discontinuous Galerkin schemes for multi-dimensional coupled hyperbolic systems

A novel class of Runge-Kutta discontinuous Galerkin schemes for coupled systems of conservation laws in multiple space dimensions that are separated by a fixed sharp interface is introduced. The schemes are derived from a relaxation approach and a local projection and do not require expensive solutions of nonlinear half-Riemann problems. The underlying Jin-Xin relaxation involves a problem specific modification of the coupling condition at the interface, for which a simple construction algorithm is presented. The schemes are endowed with higher order time discretization by means of strong stability preserving Runge-Kutta methods. These are derived from an asymptotic preserving implicit-explicit treatment of the coupled relaxation system taken to the discrete relaxation limit. In a case study the application to a multi-dimensional fluid-structure coupling problem employing the compressible Euler equations and a linear elastic model is discussed.

math.NA↗

arXiv2025

High-dimensional stochastic finite volumes using the tensor train format

A method for the uncertainty quantification of nonlinear hyperbolic conservation laws with many uncertain parameters is presented. The method combines stochastic finite volume methods and tensor trains in a novel way: the dimensions of physical space and time are kept as full tensors, while all stochastic dimensions are compressed together into a tensor train. The resulting hybrid format has one tensor train for each spatial cell and each time step. The MUSCL scheme is adapted to the proposed hybrid format, and its feasibility is demonstrated through several test cases. For the scalar Burgers' equation, we conduct a convergence study and compare the results with those obtained using the full tensor train format with three stochastic parameters. The equation is then solved for an increasing number of stochastic dimensions.For systems of conservation laws, we focus on the Euler equations. A parameter study and a comparison with the full tensor train format are carried out for the Sod shock tube problem. As a more complex application, we investigate the Shu-Osher problem, which involves intricate wave interactions. The presented method opens new avenues for integrating uncertainty quantification with established numerical schemes for hyperbolic conservation laws.

math.NA↗

arXiv2024

Numerical Boundary Control of Multi-Dimensional Hyperbolic Equations

Existing theoretical stabilization results for linear, hyperbolic multi-dimensional problems are extended to the discretized multi-dimensional problems. In contrast to existing theoretical and numerical analysis in the spatially one-dimensional case the effect of the numerical dissipation is analyzed and explicitly quantified. Further, using dimensional splitting, the numerical analysis is extended to the multi-dimensional case. The findings are confirmed by numerical simulations for low-order and high-order DG schemes both in the one-dimensional and two-dimensional case.

math.OC↗

arXiv2024

A relaxation approach to the coupling of a two-phase fluid with a linear-elastic solid

A recently developed coupling strategy for two nonconservative hyperbolic systems is employed to investigate a collapsing vapor bubble embedded in a liquid near a solid. For this purpose, an elastic solid modeled by a linear system of conservation laws is coupled to the two-phase Baer-Nunziato-type model for isothermal fluids, a nonlinear hyperbolic system with non-conservative products. For the coupling of the two systems the Jin-Xin relaxation concept is employed and embedded in a second order finite volume scheme. For a proof of concept simulations in one space dimension are performed.

math.NA↗

arXiv2024

A survey on isothermal and isentropic Baer-Nunziato-type models

Multi-component Baer-Nunziato-type models for isothermal and isentropic fluids are investigated. These are given by balance equations for volume fractions, density and momentum for each component accounting for the relaxation to equilibrium by means of relaxation terms. Mathematical properties of the models are derived such as hyperbolicity and symmetrization. The fields are characterized and corresponding Riemann invariants are determined. Appropriate entropy-entropy flux pairs are derived taking into account the phasic energy equations including the heat flux. Physically meaningful constraints are presented that ensure the entropy inequality to hold. Instantaneous relaxation to equilibrium is investigated and appropriate algorithms are presented. Numerical results for the isothermal Baer-Nunziato model are compared to an isothermal Euler model and to an isothermal phase-field model.

physics.flu-dyn↗

arXiv2024

Radially symmetric solutions of the ultra-relativistic Euler equations in several space dimensions

The ultra-relativistic Euler equations for an ideal gas are described in terms of the pressure, the spatial part of the dimensionless four-velocity and the particle density. Radially symmetric solutions of these equations are studied in two and three space dimensions. Of particular interest in the solutions are the formation of shock waves and a pressure blow up. For the investigation of these phenomena we develop a one-dimensional scheme using radial symmetry and integral conservation laws. We compare the numerical results with solutions of multi-dimensional high-order numerical schemes for general initial data in two space dimensions. The presented test cases and results may serve as interesting benchmark tests for multi-dimensional solvers.

math-ph↗

arXiv2023

Numerical schemes for coupled systems of nonconservative hyperbolic equations

A new linear relaxation system for nonconservative hyperbolic systems is introduced, in which a nonlocal source term accounts for the nonconservative product of the original system. Using an asymptotic analysis the relaxation limit and its stability are investigated. It is shown that the path-conservative Lax-Friedrichs scheme arises from a discrete limit of an implicit-explicit scheme for the relaxation system. The relaxation approach is further employed to couple two nonconservative systems at a static interface. A coupling strategy motivated from conservative Kirchhoff conditions is introduced and a corresponding Riemann solver provided. A fully discrete scheme for coupled nonconservative products is derived and studied in terms of path-conservation. Numerical experiments applying the approach to a coupled model of vascular blood flow are presented.

math.NA↗

arXiv2023

A central scheme for coupled hyperbolic systems

A novel numerical scheme to solve coupled systems of conservation laws is introduced. The scheme is derived based on a relaxation approach and does not require information on the Lax curves of the coupled systems, which simplifies the computation of suitable coupling data. The coupling condition for the underlying relaxation system plays a crucial role as it determines the behavior of the scheme in the zero relaxation limit. The role of this condition is discussed, a consistency concept with respect to the original problem is introduced, well-posedness is analyzed and explicit, nodal Riemann solvers are provided. Based on a case study considering the p-system of gas dynamics a strategy for the design of the relaxation coupling condition within the new scheme is provided.

math.NA↗

arXiv2023

Model adaptation for hyperbolic balance laws

In this work, we devise a model adaptation strategy for a class of model hierarchies consisting of two levels of model complexity. In particular, the fine model consists of a system of hyperbolic balance laws with stiff reaction terms and the coarse model consists of a system of hyperbolic conservation laws. We employ the relative entropy stability framework to obtain an a posteriori modeling error estimator. The efficiency of the model adaptation strategy is demonstrated by conducting simulations for chemically reacting fluid mixtures in one space dimension.

math.NA↗

arXiv2023

Multiresolution-analysis for stochastic hyperbolic conservation laws

A multiresolution analysis for solving stochastic conservation laws is proposed. Using a novel adaptation strategy and a higher dimensional deterministic problem, a discontinuous Galerkin (DG) solver is derived. A multiresolution analysis of the DG spaces for the proposed adaptation strategy is presented. Numerical results show that in the case of general stochastic distributions the performance of the DG solver is significantly improved by the novel adaptive strategy. The gain in efficiency is validated in computational experiments.

math.NA↗

arXiv2022

Central schemes for networked scalar conservation laws

We propose a novel scheme to numerically solve scalar conservation laws on networks without the necessity to solve Riemann problems at the junction. The scheme is derived using the relaxation system introduced in [Jin and Xin, Comm. Pure Appl. Math. 48(3), 235-276 (1995)] and taking the relaxation limit also at the nodes of the network. The scheme is mass conservative and yields well defined and easy-to-compute coupling conditions even for general networks. We discuss higher order extension of the scheme and applications to traffic flow and two-phase flow. In the former we compare with results obtained in literature.

math.NA↗

arXiv2022

Analytical investigation of 1D Darcy-Forchheimer flow under local thermal nonequilibrium

In the context of transpiration cooling, a 1D porous medium model consisting of a temperature system and a mass-momentum system is derived from the 2D/3D Darcy-Forchheimer equations. The temperatures of the coolant and the solid are assumed to be in local nonequilibrium. This system is analytically verified to have a unique solution. Transpiration cooling simulations are performed by a two-domain approach, coupling the assembled 1D porous medium solutions with 2D solutions of a hot gas flow solver. A comparison of results obtained by applying the 1D porous medium model with the temperature system either including or neglecting fluid heat conduction justifies the use of the latter simplified system.

physics.flu-dyn↗

arXiv2022

Probabilistic Constrained Bayesion Inversion for Transpiration Cooling

To enable safe operations in applications such as rocket combustion chambers, the materials require cooling to avoid material damage. Here, transpiration cooling is a promising cooling technique. Numerous studies investigate possibilities to simulate and evaluate the complex cooling mechanism. One naturally arising question is the amount of coolant required to ensure a safe operation. To study this, we introduce an approach that determines the posterior probability distribution of the Reynolds number using an inverse problem and constraining the maximum temperature of the system under parameter uncertainties. Mathematically, this chance inequality constraint is dealt with by a generalized Polynomial Chaos expansion of the system. The posterior distribution will be evaluated by different Markov Chain Monte Carlo based methods. A novel method for the constrained case is proposed and tested among others on two-dimensional transpiration cooling models.

cs.CE↗

arXiv2020

Flow Solutions of Transport Equations

Under general assumptions on the velocity field, it is possible to construct a flow that is forward untangled. Once such a flow has been selected, the associated transport problem is well-posed.

math.AP↗

arXiv2014

Transition Prediction for Scramjet Intakes Using the γ-Re_θ_t Model Coupled to Two Turbulence Models

Due to the thick boundary layers in hypersonic flows, the state of the boundary layer significantly influences the whole flow field as well as surface heat loads. Hence, for engineering applications the efficient numerical prediction of laminar-to-turbulent transition is a challenging and important task. Within the framework of the Reynolds averaged Navier-Stokes equations, Langtry/Menter [1] proposed the $γ$-$Re_{θ_t}$ transition model using two transport equations for the intermittency and $γ$-$Re_{θ_t}$ combined with the Shear Stress Transport turbulence model (SST) [2]. The transition model contains two empirical correlations for onset and length of transition. Langtry/Menter [1] designed and validated the correlations for the subsonic and transonic flow regime. For our applications in the hypersonic flow regime, the development of a new set of correlations proved necessary, even when using the same SST turbulence model [3]. Within this paper, we propose a next step and couple the transition model with the SSG/LRR-$ω$ Reynolds stress turbulence model [4] which we found to be well suited for scramjet intake simulations. First, we illustrate the necessary modifications of the Reynolds stress model and the hypersonic in-house correlations using a hypersonic flat plate test case. Next, the transition model is successfully validated for its use coupled to both turbulence models using a hypersonic double ramp test case. Regardless of the turbulence model, the transition model is able to correctly predict the transition process compared to experimental data. In addition, we apply the transition model combined with both turbulence models to three different fully 3D scramjet intake configurations which are experimentally investigated in wind tunnel facilities. The agreement with the available experimental data is also shown.

physics.flu-dyn↗

arXiv2014

Adaptive timestep control for nonstationary solutions of the Euler equations

In this paper we continue our work on adaptive timestep control for weakly non- stationary problems. The core of the method is a space-time splitting of adjoint error representations for target functionals due to S\"uli and Hartmann. The main new ingredients are (i) the extension from scalar, 1D, conservation laws to the 2D Euler equations of gas dynamics, (ii) the derivation of boundary conditions for a new formulation of the adjoint problem and (iii) the coupling of the adaptive time-stepping with spatial adaptation. For the spatial adaptation, we use a multiresolution-based strategy developed by M\"uller, and we combine this with an implicit time discretization. The combined space-time adaptive method provides an efficient choice of timesteps for implicit computations of weakly nonstationary flows. The timestep will be very large in time intervalls of stationary flow, and becomes small when a perturbation enters the flow field. The efficiency of the solver is investigated by means of an unsteady inviscid 2D flow over a bump.

math.NA↗

arXiv2013

Wavelet-based Adaptive Techniques Applied to Turbulent Hypersonic Scramjet Intake Flows

The simulation of hypersonic flows is computationally demanding due to large gradients of the flow variables caused by strong shock waves and thick boundary or shear layers. The resolution of those gradients imposes the use of extremely small cells in the respective regions. Taking turbulence into account intensives the variation in scales even more. Furthermore, hypersonic flows have been shown to be extremely grid sensitive. For the simulation of three-dimensional configurations of engineering applications, this results in a huge amount of cells and prohibitive computational time. Therefore, modern adaptive techniques can provide a gain with respect to computational costs and accuracy, allowing the generation of locally highly resolved flow regions where they are needed and retaining an otherwise smooth distribution. An h-adaptive technique based on wavelets is employed for the solution of hypersonic flows. The compressible Reynolds averaged Navier-Stokes equations are solved using a differential Reynolds stress turbulence model, well suited to predict shock-wave-boundary-layer interactions in high enthalpy flows. Two test cases are considered: a compression corner and a scramjet intake. The compression corner is a classical test case in hypersonic flow investigations because it poses a shock-wave-turbulent-boundary-layer interaction problem. The adaptive procedure is applied to a two-dimensional confguration as validation. The scramjet intake is firstly computed in two dimensions. Subsequently a three-dimensional geometry is considered. Both test cases are validated with experimental data and compared to non-adaptive computations. The results show that the use of an adaptive technique for hypersonic turbulent flows at high enthalpy conditions can strongly improve the performance in terms of memory and CPU time while at the same time maintaining the required accuracy of the results.

physics.comp-ph↗