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C5: Hybride adaptive Multiskalenmethoden für Fluide der weichen Materie

Ziel des Projektes ist es, neue hybride Multiskalenmethoden zur Simulation von Flüssigkeiten zu entwickeln, die die diskontinuierliche Galerkin (dG) Methode mit Molekulardynamik (MD) kombinieren. Die Wechselwirkungen zwischen den Teilchen- und Kontinuumsvariablen werden dynamisch bestimmt, d.h. “on the fly”. Mit Hilfe von Reduktionstechniken wird die Zahl der (rechenzeitintensiven) MD-Simulationen optimiert. In der nächsten Förderperiode soll (i) Fehlerkontrolle mit Hilfe von a priori Abschätzungen der Rückkopplung und Konvergenzanalysen studiert werden, (ii) die Methode auf komplexere physikalische Systeme wie z.B. nicht-Newtonsche Flüssigkeiten und aktive kolloidale Flüssigketen erweitert werden, und (iii) neben einfachen generischen Geometrien auch Geometrien aus typischen Mikrofluidik-Aufbauten betrachtet werden.


Convergence of finite volume schemes for the Euler equations via dissipative measure--valued solutions
E. Feireisl, M. Lukáčová-Medvid’ová, H. Mizerová
Found Comput Math 20, 923-966 (2020);
doi:10.1007/s10208-019-09433-z

The Cauchy problem for the complete Euler system is in general ill-posed in the class of admissible (entropy producing) weak solutions. This suggests that there might be sequences of approximate solutions that develop fine-scale oscillations. Accordingly, the concept of measure-valued solution that captures possible oscillations is more suitable for analysis. We study the convergence of a class of entropy stable finite volume schemes for the barotropic and complete compressible Euler equations in the multidimensional case. We establish suitable stability and consistency estimates and show that the Young measure generated by numerical solutions represents a dissipative measure-valued solution of the Euler system. Here dissipative means that a suitable form of the second law of thermodynamics is incorporated in the definition of the measure-valued solutions. In particular, using the recently established weak-strong uniqueness principle, we show that the numerical solutions converge pointwise to the regular solution of the limit systems at least on the lifespan of the latter.

A finite volume scheme for the Euler system inspired by the two velocities approach
E. Feireisl, M. Lukacova-Medvidova, H. Mizerova
Num. Math. 144 (89-132), (2020);
doi:10.1007/s00211-019-01078-y

We propose a new finite volume scheme for the Euler system of gas dynamics motivated by the model proposed by H. Brenner. Numerical viscosity imposed through upwinding acts on the velocity field rather than on the convected quantities. The resulting numerical method enjoys the crucial properties of the Euler system, in particular positivity of the approximate density and pressure and the minimal entropy principle. In addition, the approximate solutions generate a dissipative measure-valued solutions of the limit system. In particular, the numerical solutions converge to the smooth solution of the system as long as the latter exists.

K-convergence as a new tool in numerical analysis
E.Feireisl, M. Lukacova-Medvidova, H. Mizerova
IMA J. Num. Anal. 40, 2227–2255 (2020);
doi:10.1093/imanum/drz045

We adapt the concept of K-convergence of Young measures to the sequences of approximate solutions resulting from numerical schemes. We obtain new results on pointwise convergence of numerical solutions in the case when solutions of the limit continuous problem possess minimal regularity. We apply the abstract theory to a finite volume method for the isentropic Euler system describing the motion of a compressible inviscid fluid. The result can be seen as a nonlinear version of the fundamental Lax equivalence theorem.

On the convergence of a finite volume method for the Navier–Stokes–Fourier system
E.Feireisl, M. Lukacova-Medvidova, H. Mizerova, B. She
IMA J. Num. Anal. , (2020);
C5 Project
doi:10.1093/imanum/draa060

We study convergence of a finite volume scheme for the Navier-Stokes-Fourier system describing the motion of compressible viscous and heat conducting fluids. The numerical flux uses upwinding with an additional numerical diffusion of order O(h^{ε+1}), 0<ε<1. The approximate solutions are piecewise constant functions with respect to the underlying mesh. We show that any uniformly bounded sequence of numerical solutions converges unconditionally to the solution of the Navier-Stokes-Fourier system. In particular, the existence of the solution to the Navier-Stokes-Fourier system is not a priori assumed.

Convergence of a finite volume scheme for the compressible Navier-Stokes system
E.Feireisl, M. Lukacova-Medvidova, H. Mizerova
ESAIM: Math. Model. Num. 53, 1957–1979 (2019);
doi: https://doi.org/10.1051/m2an/2019043

We study convergence of a finite volume scheme for the compressible (barotropic) Navier–Stokes system. First we prove the energy stability and consistency of the scheme and show that the numerical solutions generate a dissipative measure-valued solution of the system. Then by the dissipative measure-valued-strong uniqueness principle, we conclude the convergence of the numerical solution to the strong solution as long as the latter exists. Numerical experiments for standard benchmark tests support our theoretical results.

An asymptotic preserving scheme for kinetic chemotaxis models in two space dimensions
A. Chertock, A. Kurganov, M. Lukacova-Medvidova, S. Nur Oezcan
Kinetic and Related Models 12 (1), 195–216 (2019);
URL: http://aimsciences.org//article/doi/10.3934/krm.2019009
doi:10.3934/krm.2019009

In this paper, we study two-dimensional multiscale chemotaxis models based on a combination of the macroscopic evolution equation for chemoattractant and microscopic models for cell evolution. The latter is governed by a Boltzmann-type kinetic equation with a local turning kernel operator which describes the velocity change of the cells. The parabolic scaling yields a non-dimensional kinetic model with a small parameter, which represents the mean free path of the cells. We propose a new asymptotic preserving numerical scheme that reflects the convergence of the studied micro-macro model to its macroscopic counterpart-the Patlak-Keller-Segel system-in the singular limit. The method is based on the operator splitting strategy and a suitable combination of the higher-order implicit and explicit time discretizations. In particular, we use the so-called even-odd decoupling and approximate the stiff terms arising in the singular limit implicitly. We prove that the resulting scheme satisfies the asymptotic preserving property. More precisely, it yields a consistent approximation of the Patlak-Keller-Segel system as the mean-free path tends to 0. The derived asymptotic preserving method is used to get better insight to the blowup behavior of two-dimensional kinetic chemotaxis model.

Convergence of a mixed finite element finite volume scheme for the isentropic Navier-Stokes system via dissipative measure-valued solutions
E. Feireisl, M. Lukacova-Medvidova
Found. Comput. Math. 18 , 703–730 (2018);
doi: DOI: 10.1007/s10208-017-9351-2

We study convergence of a mixed finite element-finite volume numerical scheme for the isentropic Navier-Stokes system under the full range of the adiabatic exponent. We establish suitable stability and consistency estimates and show that the Young measure generated by numerical solutions represents a dissipative measure-valued solutions of the limit system. In particular, using the recently established weak{strong uniqueness principle in the class of dissipative measure-valued solutions we show that the numerical solutions converge strongly to a strong solutions of the limit system as long as the latter exists.

Asymptotic preserving error estimates for numerical solutions of compressible Navier-Stokes equations in the low Mach number regime
E. Feireisl, M. Lukacova-Medvidova, S. Necasova, A. Novotny, B. She
SIAM Multiscale Model. Simul. 16 (1), 150–183 (2018);
URL: https://epubs.siam.org/doi/10.1137/16M1094233

We study the convergence of numerical solutions of the compressible Navier-Stokes system to its incompressible limit. The numerical solution is obtained by a combined finite element-finite volume method based on the linear Crouzeix-Raviart finite element for the velocity and piecewise constant approximation for the density. The convective terms are approximated using upwinding. The distance between a numerical solution of the compressible problem and the strong solution of the incompressible Navier-Stokes equations is measured by means of a relative energy functional. For barotropic pressure exponent larger than 3/2 and for well-prepared initial data we obtain uniform convergence of order. Extensive numerical simulations confirm that the numerical solution of the compressible problem converges to the solution of the incompressible Navier-Stokes equations as the discretization parameters and the Mach number tend to zero.

Molecular dynamics simulations in hybrid particle-continuum schemes: Pitfalls and caveats
S. Stalter, L. Yelash, N. Emamy, A. Statt, M. Hanke, M. Lukáčová-Medvid’ová, P. Virnau
Computer Physics Communications, (2017);
doi:10.1016/j.cpc.2017.10.016

Asymptotic preserving IMEX finite volume schemes for low Mach number Euler equations with gravitation
G. Bispen, M. Lukacova-Medvidova, L. Yelash
J. Comput. Phys. 335, 222-248 (2017);
doi:10.1016/j.jcp.2017.01.020

Reduced-order hybrid multiscale method combining the molecular dynamics and the discontinuous Galerkin method
N. Emamy, M. Lukácová-Medvid’ová, S. Stalter, P. Virnau, L. Yelash
VII International Conference on Computational Methods for Coupled Problems in Science and Engineering, Coupled Problems 2017, 1-15. (2017);
URL: http://congress.cimne.com/coupled2017/frontal/default.asp

We present a new reduced-order hybrid multiscale method to simulate complex fluids. The method combines the continuum and molecular descriptions. We follow the framework of the heterogeneous multi-scale method (HMM) that makes use of the scale separation into macro- and micro-levels. On the macro-level, the governing equations of the incompressible flow are the continuity and momentum equations. The equations are solved using a high-order accurate discontinuous Galerkin Finite Element Method (dG) and implemented in the BoSSS code. The missing information on the macro-level is represented by the unknown stress tensor evaluated by means of the molecular dynamics (MD) simulations on the micro-level. We shear the microscopic system by applying Lees-Edwards boundary conditions and either an isokinetic or Lowe-Andersen thermostat. The data obtained from the MD simulations underlie large stochastic errors that can be controlled by means of the least-square approximation. In order to reduce a large number of computationally expensive MD runs, we apply the reduced order approach. Numerical experiments confirm the robustness of our newly developed hybrid MD-dG method.

Analysis and numerical solution of the Peterlin viscoelastic model (Doktorarbeit)
Mizerova Hana
JGU (2015);
URL: http://ubm.opus.hbz-nrw.de/volltexte/2015/4231/

Liquids and gasses form a vital part of nature. Many of these are complex fluids with non-Newtonian behaviour. We introduce a mathematical model describing the unsteady motion of an incompressible polymeric fluid. Each polymer molecule is treated as two beads connected by a spring. For the nonlinear spring force it is not possible to obtain a closed system of equations, unless we approximate the force law. The Peterlin approximation replaces the length of the spring by the length of the average spring. Consequently, the macroscopic dumbbell-based model for dilute polymer solutions is obtained. The model consists of the conservation of mass and momentum and time evolution of the symmetric positive definite conformation tensor, where the diffusive effects are taken into account. In two space dimensions we prove global in time existence of weak solutions. Assuming more regular data we show higher regularity and consequently uniqueness of the weak solution. For the Oseen-type Peterlin model we propose a linear pressure-stabilized characteristics finite element scheme. We derive the corresponding error estimates and we prove, for linear finite elements, the optimal first order accuracy. Theoretical error of the pressure-stabilized characteristic finite element scheme is confirmed by a series of numerical experiments.

Accelerated GPU simulation of compressible flow by the discontinuous evolution Galerkin method
B. J. Block, M. Lukáčová-Medvid’ová, P. Virnau, L. Yelash
The European Physical Journal Special Topics 210 (1), 119-132 (2012);
doi:10.1140/epjst/e2012-01641-0

The aim of the present paper is to report on our recent results for GPU accelerated simulations of compressible flows. For numerical simulation the adaptive discontinuous Galerkin method with the multidimensional bicharacteristic based evolution Galerkin operator has been used. For time discretization we have applied the explicit third order Runge-Kutta method. Evaluation of the genuinely multidimensional evolution operator has been accelerated using the GPU implementation. We have obtained a speedup up to 30 (in comparison to a single CPU core) for the calculation of the evolution Galerkin operator on a typical discretization mesh consisting of 16384 mesh cells.

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