arXiv subjects

Christian Rohde

Publications and source records attributed to Christian Rohde.

67 records · Page 4Linked to original sources

arXiv2019

A posteriori error analysis and adaptive non-intrusive numerical schemes for systems of random conservation laws

In this article we consider one-dimensional random systems of hyperbolic conservation laws. We first establish existence and uniqueness of random entropy admissible solutions for initial value problems of conservation laws which involve random initial data and random flux functions. Based on these results we present an a posteriori error analysis for a numerical approximation of the random entropy admissible solution. For the stochastic discretization, we consider a non-intrusive approach, the Stochastic Collocation method. The spatio-temporal discretization relies on the Runge--Kutta Discontinuous Galerkin method. We derive the a posteriori estimator using continuous 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. We conclude with various numerical examples investigating the scaling properties of the residuals and illustrating the efficiency of the proposed adaptive algorithm.

math.NA↗

arXiv2018

Existence of weak solutions for a nonlocal pseudo-parabolic model for Brinkman two-phase flow in asymptotically flat porous media

We study a nonlocal evolution equation that involves a pseudo-parabolic third-order term. The equation models almost uni-directional two-phase flow in Brinkman regimes. We prove the existence of weak solutions for this equation. We also give a series of numerical examples that demonstrate the ability of the equation to support overshooting and explore the behavior of solutions in various limit regimes.

math.AP↗

arXiv2018

A posteriori error analysis for random scalar conservation laws using the Stochastic Galerkin method

In this article we present an a posteriori error estimator for the spatial-stochastic error of a Galerkin-type discretisation of an initial value problem for a random hyperbolic conservation law. For the stochastic discretisation we use the Stochastic Galerkin method and for the spatial-temporal discretisation of the Stochastic Galerkin system a Runge-Kutta Discontinuous Galerkin method. The estimator is obtained using smooth reconstructions of the discrete solution. Combined with the relative entropy stability framework of Dafermos \cite{dafermos2005hyperbolic}, this leads to computable error bounds for the space-stochastic discretisation error. Moreover, it turns out that the error estimator admits a splitting into one part representing the spatial error, and a remaining term, which can be interpreted as the stochastic error. This decomposition allows us to balance the errors arising from spatial and stochastic discretisation. We conclude with some numerical examples confirming the theoretical findings.

math.NA↗

arXiv2018

A Particle-based Multiscale Solver for Compressible Liquid-Vapor Flow

To describe complex flow systems accurately, it is in many cases important to account for the properties of fluid flows on a microscopic scale. In this work, we focus on the description of liquid-vapor flow with a sharp interface between the phases. The local phase dynamics at the interface can be interpreted as a Riemann problem for which we develop a multiscale solver in the spirit of the heterogeneous multiscale method, using a particle-based microscale model to augment the macroscopic two-phase flow system. The application of a microscale model makes it possible to use the intrinsic properties of the fluid at the microscale, instead of formulating (ad-hoc) constitutive relations.

math.NA↗

arXiv2018

Comparison of data-driven uncertainty quantification methods for a carbon dioxide storage benchmark scenario

A variety of methods is available to quantify uncertainties arising with\-in the modeling of flow and transport in carbon dioxide storage, but there is a lack of thorough comparisons. Usually, raw data from such storage sites can hardly be described by theoretical statistical distributions since only very limited data is available. Hence, exact information on distribution shapes for all uncertain parameters is very rare in realistic applications. We discuss and compare four different methods tested for data-driven uncertainty quantification based on a benchmark scenario of carbon dioxide storage. In the benchmark, for which we provide data and code, carbon dioxide is injected into a saline aquifer modeled by the nonlinear capillarity-free fractional flow formulation for two incompressible fluid phases, namely carbon dioxide and brine. To cover different aspects of uncertainty quantification, we incorporate various sources of uncertainty such as uncertainty of boundary conditions, of conceptual model definitions and of material properties. We consider recent versions of the following non-intrusive and intrusive uncertainty quantification methods: arbitary polynomial chaos, spatially adaptive sparse grids, kernel-based greedy interpolation and hybrid stochastic Galerkin. The performance of each approach is demonstrated assessing expectation value and standard deviation of the carbon dioxide saturation against a reference statistic based on Monte Carlo sampling. We compare the convergence of all methods reporting on accuracy with respect to the number of model runs and resolution. Finally we offer suggestions about the methods' advantages and disadvantages that can guide the modeler for uncertainty quantification in carbon dioxide storage and beyond.

cs.CE↗

arXiv2017

On Darcy-and Brinkman-Type Models for Two-Phase Flow in Asymptotically Flat Domains

We study two-phase flow for Darcy and Brinkman regimes. To reduce the computational complexity for flow in vertical equilibrium various simplified models have been suggested. Examples are dimensional reduction by vertical integration, the multiscale model approach in [Guo et al., 2014] or the asymptotic approach in [Yortsos, 1995]. The latter approach uses a geometrical scaling. We show the efficiency of the approach in asymptotically flat domains for Darcy regimes. Moreover, we prove that it is vastly equivalent to the multiscale model approach. We apply then asymptotic analysis to the two-phase flow model in Brinkman regimes. The limit model is a single nonlocal evolution law with a pseudo-parabolic extension. Its computational efficiency is demonstrated using numerical examples. Finally, we show that the new limit model exhibits overshoot behaviour as it has been observed for dynamical capillarity laws [Hassanizadeh and Gray, 1993].

math.AP↗

arXiv2017

A linear domain decomposition method for two-phase flow in porous media

This article is a follow up of our submitted paper [11] in which a decomposition of the Richards equation along two soil layers was discussed. A decomposed problem was formulated and a decoupling and linearisation technique was presented to solve the problem in each time step in a fixed point type iteration. This article extends these ideas to the case of two-phase in porous media and the convergence of the proposed domain decomposition method is rigorously shown.

math.NA↗

arXiv2017

A linear domain decomposition method for partially saturated flow in porous media

The Richards equation is a nonlinear parabolic equation that is commonly used for modelling saturated/unsaturated flow in porous media. We assume that the medium occupies a bounded Lipschitz domain partitioned into two disjoint subdomains separated by a fixed interface $Γ$. This leads to two problems defined on the subdomains which are coupled through conditions expressing flux and pressure continuity at $Γ$. After an Euler implicit discretisation of the resulting nonlinear subproblems a linear iterative ($L$-type) domain decomposition scheme is proposed. The convergence of the scheme is proved rigorously. In the last part we present numerical results that are in line with the theoretical finding, in particular the unconditional convergence of the scheme. We further compare the scheme to other approaches not making use of a domain decomposition. Namely, we compare to a Newton and a Picard scheme. We show that the proposed scheme is more stable than the Newton scheme while remaining comparable in computational time, even if no parallelisation is being adopted. Finally we present a parametric study that can be used to optimize the proposed scheme.

math.NA↗

arXiv2016

On Riemann Solvers and Kinetic Relations for Isothermal Two-Phase Flows with Surface Tension

We consider a sharp-interface approach for the inviscid isothermal dynamics of compressible two-phase flow, that accounts for phase transition and surface tension effects. To fix the mass exchange and entropy dissipation rate across the interface kinetic relations are frequently used. The complete uni-directional dynamics can then be understood by solving generalized two-phase Riemann problems. We present new well-posedness theorems for the Riemann problem and corresponding computable Riemann solvers, that cover quite general equations of state, metastable input data and curvature effects. The new Riemann solver is used to validate different kinetic relations on physically relevant problems including a comparison with experimental data. Riemann solvers are building blocks for many numerical schemes that are used to track interfaces in two-phase flow. It is shown that the new Riemann solver enables reliable and efficient computations for physical situations that could not be treated before.

physics.flu-dyn↗

arXiv2016

Effect of surfactant concentration and interfacial slip on the flow past a viscous drop at low surface Péclet number

The motion of a viscous drop is investigated when the interface is fully covered with a stagnant layer of surfactant in an arbitrary unsteady Stokes flow for the low surface Péclet number limit. The effect of the interfacial slip coefficient on the behavior of the flow field is also considered. The hydrodynamic problem is solved by the solenoidal decomposition method and the drag force is computed in terms of Faxen's laws using a perturbation ansatz in powers of the surface Péclet number. The analytical expressions for the migration velocity of the drop are also obtained in powers of the surface Péclet number. Further instances corresponding to a given ambient flow as uniform flow, Couette flow, Poiseuille flow are analyzed. Moreover, it is observed that, a surfactant-induced cross-stream migration of the drop occur towards the centre-line in both Couette flow and Poiseuille flow cases. The variation of the drag force and migration velocity is computed for different parameters such as Péclet number, Marangoni number etc.

physics.flu-dyn↗

arXiv2015

A sharp interface method for compressible liquid-vapor flow with phase transition and surface tension

The numerical approximation of non-isothermal liquid-vapor flow within the compressible regime is a difficult task because complex physical effects at the phase interfaces can govern the global flow behavior. We present a sharp interface approach which treats the interface as a shock-wave like discontinuity. Any mixing of fluid phases is avoided by using the flow solver in the bulk regions only, and a ghost-fluid approach close to the interface. The coupling states for the numerical solution in the bulk regions are determined by the solution of local multi-phase Riemann problems across the interface. The Riemann solution accounts for the relevant physics by enforcing appropriate jump conditions at the phase boundary. A wide variety of interface effects can be handled in a thermodynamically consistent way. This includes surface tension or mass/energy transfer by phase transition. Moreover, the local normal speed of the interface, which is needed to calculate the time evolution of the interface, is given by the Riemann solution. The interface tracking itself is based on a level-set method. The focus in this paper is the description of the multi-phase Riemann solver and its usage within the sharp interface approach. One-dimensional problems are selected to validate the approach. Finally, the three-dimensional simulation of a wobbling droplet and a shock droplet interaction in two dimensions are shown. In both problems phase transition and surface tension determine the global bulk behavior.

physics.flu-dyn↗

arXiv2015

Energy-Efficient 5G Outdoor-to-Indoor Communication: SUDAS Over Licensed and Unlicensed Spectrum

In this paper, we study the joint resource allocation algorithm design for downlink and uplink multicarrier transmission assisted by a shared user equipment (UE)-side distributed antenna system (SUDAS). The proposed SUDAS simultaneously utilizes licensed frequency bands and unlicensed frequency bands, (e.g. millimeter wave bands), to enable a spatial multiplexing gain for single-antenna UEs to improve energy efficiency and system throughput of $5$-th generation (5G) outdoor-to-indoor communication. The design of the UE selection, the time allocation to uplink and downlink, and the transceiver processing matrix is formulated as a non-convex optimization problem for the maximization of the end-to-end system energy efficiency (bits/Joule). The proposed problem formulation takes into account minimum data rate requirements for delay sensitive UEs and the circuit power consumption of all transceivers. In order to design a tractable resource allocation algorithm, we first show that the optimal transmitter precoding and receiver post-processing matrices jointly diagonalize the end-to-end communication channel for both downlink and uplink communication via SUDAS. Subsequently, the matrix optimization problem is converted to an equivalent scalar optimization problem for multiple parallel channels, which is solved by an asymptotically globally optimal iterative algorithm. Besides, we propose a suboptimal algorithm which finds a locally optimal solution of the non-convex optimization problem. Simulation results illustrate that the proposed resource allocation algorithms for SUDAS achieve a significant performance gain in terms of system energy efficiency and spectral efficiency compared to conventional baseline systems by offering multiple parallel data streams for single-antenna UEs.

cs.IT↗

arXiv2015

Resource Allocation for Outdoor-to-Indoor Multicarrier Transmission with Shared UE-side Distributed Antenna Systems

In this paper, we study the resource allocation algorithm design for downlink multicarrier transmission with a shared user equipment (UE)-side distributed antenna system (SUDAS) which utilizes both licensed and unlicensed frequency bands for improving the system throughput. The joint UE selection and transceiver processing matrix design is formulated as a non-convex optimization problem for the maximization of the end-to-end system throughput (bits/s). In order to obtain a tractable resource allocation algorithm, we first show that the optimal transmitter precoding and receiver post-processing matrices jointly diagonalize the end-to-end communication channel. Subsequently, the optimization problem is converted to a scalar optimization problem for multiple parallel channels, which is solved by using an asymptotically optimal iterative algorithm. Simulation results illustrate that the proposed resource allocation algorithm for the SUDAS achieves an excellent system performance and provides a spatial multiplexing gain for single-antenna UEs.

cs.IT↗