scholarly journals Stochastic dynamics for inextensible fibers in a spatially semi-discrete setting

2017 ◽  
Vol 17 (02) ◽  
pp. 1750016 ◽  
Author(s):  
Felix Lindner ◽  
Nicole Marheineke ◽  
Holger Stroot ◽  
Alexander Vibe ◽  
Raimund Wegener
Keyword(s):  
Turbulent Flows ◽  
Lagrange Multiplier ◽  
Stochastic Dynamics ◽  
Time Discretization ◽  
Explicit Representation ◽  
Beam Model ◽  
Time Step ◽  
Continuous Space ◽  
Linear Growth Condition ◽  
Inextensible Fibers

We investigate a spatially discrete surrogate model for the dynamics of a slender, elastic, inextensible fiber in turbulent flows. Deduced from a continuous space-time beam model for which no solution theory is available, it consists of a high-dimensional second order stochastic differential equation in time with a nonlinear algebraic constraint and an associated Lagrange multiplier term. We establish a suitable framework for the rigorous formulation and analysis of the semi-discrete model and prove existence and uniqueness of a global strong solution. The proof is based on an explicit representation of the Lagrange multiplier and on the observation that the obtained explicit drift term in the equation satisfies a one-sided linear growth condition on the constraint manifold. The theoretical analysis is complemented by numerical studies concerning the time discretization of our model. The performance of implicit Euler-type methods can be improved when using the explicit representation of the Lagrange multiplier to compute refined initial estimates for the Newton method applied in each time step.

GLOBAL EXISTENCE FOR PHASE TRANSITION PROBLEMS VIA A VARIATIONAL SCHEME

2004 ◽  
Vol 01 (04) ◽  
pp. 747-768
Author(s):  
CHRISTIAN ROHDE ◽  
MAI DUC THANH
Keyword(s):  
Phase Transition ◽  
Surface Tension ◽  
Initial Data ◽  
Time Discretization ◽  
Approximate Solutions ◽  
Implicit Time ◽  
Time Step ◽  
Dynamical Phase Transition ◽  
Transition Dynamics ◽  
Variational Scheme

We construct approximate solutions of the initial value problem for dynamical phase transition problems via a variational scheme in one space dimension. First, we deal with a local model of phase transition dynamics which contains second and third order spatial derivatives modeling the effects of viscosity and surface tension. Assuming that the initial data are periodic, we prove the convergence of approximate solutions to a weak solution which satisfies the natural dissipation inequality. We note that this result still holds for non-periodic initial data. Second, we consider a model of phase transition dynamics with only Lipschitz continuous stress–strain function which contains a non-local convolution term to take account of surface tension. We also establish the existence of weak solutions. In both cases the proof relies on implicit time discretization and the analysis of a minimization problem at each time step.

Continuous Galerkin Petrov Time Discretization Scheme for the Solutions of the Chen System

2015 ◽  
Vol 10 (6) ◽  
Author(s):  
S. Hussain ◽  
Z. Salleh
Keyword(s):  
Fourth Order ◽  
Three Dimensional ◽  
Time Discretization ◽  
Dimensional System ◽  
Numerical Comparison ◽  
Chen System ◽  
Time Step ◽  
Quadratic Nonlinearities ◽  
Continuous Galerkin ◽  
Three Dimensional System

In this paper, the continuous Galerkin Petrov time discretization (cGP) scheme is applied to the Chen system, which is a three-dimensional system of ordinary differential equations (ODEs) with quadratic nonlinearities. In particular, we implement and analyze numerically the higher order cGP(2)-method which is found to be of fourth order at the discrete time points. A numerical comparison with classical fourth-order Runge–Kutta (RK4) is given for the presented problem. We look at the accuracy of the cGP(2) as the Chen system changes from a nonchaotic system to a chaotic one. It is shown that the cGP(2) method gains accurate results at larger time step sizes for both cases.

Direct Simulation of Electrorheological Suspensions Subjected to Spatially Nonuniform Electric Field

2002 ◽  
Author(s):  
J. Kadaksham ◽  
P. Singh ◽  
N. Aubry
Keyword(s):  
Electric Field ◽  
Electric Fields ◽  
Lagrange Multiplier ◽  
Dipole Approximation ◽  
Nonuniform Electric Field ◽  
Lagrange Multiplier Method ◽  
Body Motion ◽  
Multiplier Method ◽  
Point Dipole ◽  
Time Step

A numerical method based on the distributed Lagrange Multiplier method (DLM) [2,8] is developed for direct simulations of electrorheological (ER) liquids subjected to spatially varying electric fields. The flow inside particle boundaries is constrained to be rigid body motion by the distributed Lagrange multiplier method. The point-dipole approximation [6] is used to model the electrostatic forces acting on the polarized particles. The code is verified by performing a convergence study that shows that the results are independent of mesh and time step sizes. In a spatially nonuniform electric field the particles move to the regions where the magnitude of electric field is locally maximum when the particle permittivity is greater than that of the liquid. On the other hand, when the particle permittivity is smaller than that of the liquid the particles move to the regions of local minimum of electric field.

Fully discrete numerical schemes of a data assimilation algorithm: uniform-in-time error estimates

2019 ◽  
Vol 40 (4) ◽  
pp. 2584-2625 ◽  
Author(s):  
Hussain A Ibdah ◽  
Cecilia F Mondaini ◽  
Edriss S Titi
Keyword(s):  
Data Assimilation ◽  
Error Estimates ◽  
Stokes Equations ◽  
Time Discretization ◽  
Dissipative Systems ◽  
Complete Analysis ◽  
Two Dimensional ◽  
Time Error ◽  
Time Step ◽  
Fully Discrete

Abstract Our aim is to approximate a reference velocity field solving the two-dimensional Navier–Stokes equations (NSE) in the absence of its initial condition by utilizing spatially discrete measurements of that field, available at a coarse scale, and continuous in time. The approximation is obtained via numerically discretizing a downscaling data assimilation algorithm. Time discretization is based on semiimplicit and fully implicit Euler schemes, while spatial discretization (which can be done at an arbitrary scale regardless of the spatial resolution of the measurements) is based on a spectral Galerkin method. The two fully discrete algorithms are shown to be unconditionally stable, with respect to the size of the time step, the number of time steps and the number of Galerkin modes. Moreover, explicit, uniform-in-time error estimates between the approximation and the reference solution are obtained, in both the $L^2$ and $H^1$ norms. Notably, the two-dimensional NSE, subject to the no-slip Dirichlet or periodic boundary conditions, are used in this work as a paradigm. The complete analysis that is presented here can be extended to other two- and three-dimensional dissipative systems under the assumption of global existence and uniqueness.

scholarly journals An entropy minimization approach to second-order variational mean-field games

2019 ◽  
Vol 29 (08) ◽  
pp. 1553-1583 ◽  
Author(s):  
Jean-David Benamou ◽  
Guillaume Carlier ◽  
Simone Di Marino ◽  
Luca Nenna
Keyword(s):  
Efficient Algorithm ◽  
Mean Field ◽  
Time Discretization ◽  
Second Order ◽  
Mean Field Games ◽  
Time Step ◽  
Entropy Minimization ◽  
Convergence Results ◽  
Quadratic Hamiltonian ◽  
Minimization Approach

We propose an entropy minimization viewpoint on variational mean-field games with diffusion and quadratic Hamiltonian. We carefully analyze the time discretization of such problems, establish [Formula: see text]-convergence results as the time step vanishes and propose an efficient algorithm relying on this entropic interpretation as well as on the Sinkhorn scaling algorithm.

scholarly journals Assessment of Hydrometeor Collection Rates from Exact and Approximate Equations. Part II: Numerical Bounding

2007 ◽  
Vol 46 (1) ◽  
pp. 82-96
Author(s):  
Brian J. Gaudet ◽  
Jerome M. Schmidt
Keyword(s):  
Mass Transfer ◽  
Exact Solution ◽  
Splitting Method ◽  
Time Discretization ◽  
Large Mass ◽  
Time Step ◽  
Transfer Rates ◽  
Mass Moment ◽  
Time Splitting ◽  
Approximate Equations

Abstract Past microphysical investigations, including Part I of this study, have noted that the collection equation, when applied to the interaction between different hydrometeor species, can predict large mass transfer rates, even when an exact solution is used. The fractional depletion in a time step can even exceed unity for the collected species with plausible microphysical conditions and time steps, requiring “normalization” by a microphysical scheme. Although some of this problem can be alleviated through the use of more moment predictions and hydrometeor categories, the question as to why such “overdepletion” can be predicted in the first place remains insufficiently addressed. It is shown through both physical and conceptual arguments that the explicit time discretization of the bulk collection equation for any moment is not consistent with a quasi-stochastic view of collection. The result, under certain reasonable conditions, is a systematic overprediction of collection, which can become a serious error in the prediction of higher-order moments in a bulk scheme. The term numerical bounding is used to refer to the use of a quasi-stochastically consistent formula that prevents fractional collections exceeding unity for any moments. Through examples and analysis it is found that numerical bounding is typically important in mass moment prediction for time steps exceeding approximately 10 s. The Poisson-based numerical bounding scheme is shown to be simple in application and conceptualization; within a straightforward idealization it completely corrects overdepletion while even being immune to the rediagnosis error of the time-splitting method. The scheme’s range of applicability and utility are discussed.

A Fast Semi-Implicit Level Set Method for Curvature Dependent Flows with an Application to Limit Cycles Extraction in Dynamical Systems

2015 ◽  
Vol 18 (1) ◽  
pp. 203-229 ◽  
Author(s):  
Guoqiao You ◽  
Shingyu Leung
Keyword(s):  
Dynamical Systems ◽  
Level Set Method ◽  
Level Set ◽  
Limit Cycles ◽  
Time Discretization ◽  
Numerical Approach ◽  
Initial Guess ◽  
Implicit Time ◽  
Time Step ◽  
Image Regularization

AbstractWe propose a new semi-implicit level set approach to a class of curvature dependent flows. The method generalizes a recent algorithm proposed for the motion by mean curvature where the interface is updated by solving the Rudin-Osher-Fatemi (ROF) model for image regularization. Our proposal is general enough so that one can easily extend and apply the method to other curvature dependent motions. Since the derivation is based on a semi-implicit time discretization, this suggests that the numerical scheme is stable even using a time-step significantly larger than that of the corresponding explicit method. As an interesting application of the numerical approach, we propose a new variational approach for extracting limit cycles in dynamical systems. The resulting algorithm can automatically detect multiple limit cycles staying inside the initial guess with no condition imposed on the number nor the location of the limit cycles. Further, we also propose in this work an Eulerian approach based on the level set method to test if the limit cycles are stable or unstable.

NUMERICAL GRADIENT FLOW DISCRETIZATION OF VISCOUS THIN FILMS ON CURVED GEOMETRIES

2013 ◽  
Vol 23 (05) ◽  
pp. 917-947 ◽  
Author(s):  
MARTIN RUMPF ◽  
ORESTIS VANTZOS
Keyword(s):  
Thin Films ◽  
Transport Equation ◽  
Degrees Of Freedom ◽  
Gradient Flow ◽  
Time Discretization ◽  
Time Step ◽  
Time Stepping ◽  
Exterior Calculus ◽  
Natural Time ◽  
Space Discretization

The evolution of a viscous thin film on a curved geometry is numerically approximated based on the natural time discretization of the underlying gradient flow. This discretization leads to a variational problem to be solved at each time step, which reflects the balance between the decay of the free (gravitational and surface) energy and the viscous dissipation. Both dissipation and energy are derived from a lubrication approximation for a small ratio between the characteristic film height and the characteristic length scale of the surface. The dissipation is formulated in terms of a corresponding flux field, whereas the energy primarily depends on the fluid volume per unit surface, which is a conserved quantity. These two degrees of freedom are coupled by the underlying transport equation. Hence, one is naturally led to a PDE-constrained optimization problem, where the variational time stepping problem has to be solved under the constraint described by the transport equation. For the space discretization a discrete exterior calculus approach is investigated. Various applications demonstrate the qualitative and quantitative behavior of one- and two-dimensional thin films on curved geometries.

scholarly journals Stability under Galerkin truncation of A-stable Runge–Kutta discretizations in time

2014 ◽  
Vol 144 (3) ◽  
pp. 603-636 ◽  
Author(s):  
Marcel Oliver ◽  
Claudia Wulff
Keyword(s):  
Spatial Resolution ◽  
Truncation Error ◽  
Evolution Equations ◽  
Linear Part ◽  
Time Discretization ◽  
Time Step ◽  
Nonlinear Part ◽  
Scale Of Hilbert Spaces ◽  
Galerkin Truncation ◽  
Runge Kutta

We consider semilinear evolution equations for which the linear part is normal and generates a strongly continuous semigroup, and the nonlinear part is sufficiently smooth on a scale of Hilbert spaces. We approximate their semi-flow by an implicit A-stable Runge–Kutta discretization in time and a spectral Galerkin truncation in space. We show regularity of the Galerkin-truncated semi-flow and its time discretization on open sets of initial values with bounds that are uniform in the spatial resolution and the initial value. We also prove convergence of the space-time discretization without any condition that couples the time step to the spatial resolution. We then estimate the Galerkin truncation error for the semi-flow of the evolution equation, its Runge–Kutta discretization and their respective derivatives, showing how the order of the Galerkin truncation error depends on the smoothness of the initial data. Our results apply, in particular, to the semilinear wave equation and to the nonlinear Schrodinger equation.

scholarly journals A Lie group variational integration approach to the full discretization of a constrained geometrically exact Cosserat beam model

2021 ◽  
Author(s):  
Stefan Hante ◽  
Denise Tumiotto ◽  
Martin Arnold
Keyword(s):  
Lie Group ◽  
Group Structure ◽  
Equations Of Motion ◽  
Second Order ◽  
The Body ◽  
Benchmark Problems ◽  
Beam Model ◽  
Step Size ◽  
Time Step ◽  
Time Step Size

AbstractIn this paper, we will consider a geometrically exact Cosserat beam model taking into account the industrial challenges. The beam is represented by a framed curve, which we parametrize in the configuration space $\mathbb{S}^{3}\ltimes \mathbb{R}^{3}$ S 3 ⋉ R 3 with semi-direct product Lie group structure, where $\mathbb{S}^{3}$ S 3 is the set of unit quaternions. Velocities and angular velocities with respect to the body-fixed frame are given as the velocity vector of the configuration. We introduce internal constraints, where the rigid cross sections have to remain perpendicular to the center line to reduce the full Cosserat beam model to a Kirchhoff beam model. We derive the equations of motion by Hamilton’s principle with an augmented Lagrangian. In order to fully discretize the beam model in space and time, we only consider piecewise interpolated configurations in the variational principle. This leads, after approximating the action integral with second order, to the discrete equations of motion. Here, it is notable that we allow the Lagrange multipliers to be discontinuous in time in order to respect the derivatives of the constraint equations, also known as hidden constraints. In the last part, we will test our numerical scheme on two benchmark problems that show that there is no shear locking observable in the discretized beam model and that the errors are observed to decrease with second order with the spatial step size and the time step size.

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