Time Discretization Parameters Research Articles

Fully discrete schemes for monotone optimal control problems

In this article, we study an infinite horizon optimal control problem with monotone controls. We analyze the associated Hamilton–Jacobi–Bellman (HJB) variational inequality which characterizes the value function and consider the totally discretized problem using Lagrange elements to approximate the state space \(\Omega \). The convergence orders of these approximations are proved, which are in general \((h+\frac{k}{\sqrt{h}})^\gamma \) where \(\gamma \) is the Holder constant of the value function u, h and k are the time and space discretization parameters, respectively. A special election of the relations between h and k allows to obtain a convergence of order \(k^{\frac{2}{3}\gamma }\), which is valid without semiconcavity hypotheses over the problem’s data.

Influence of Time Discretization and Input Parameter on the ANN Based Synthetic Streamflow Generation

The capability of ANN to generate synthetic series of river discharge averaged over different time steps with limited data has been investigated in the present study. While an ANN model with certain input parameters can generate a monthly averaged streamflow series efficiently; it fails to generate a series of smaller time steps with the same accuracy. The scope of improving efficiency of ANN in generating synthetic streamflow by using different combinations of input data has been analyzed. The developed models have been assessed through their application in the river Subansiri in India. Efficiency of the ANN models has been evaluated by comparing ANN generated series with the historical series and the series generated by Thomas-Fiering model on the basis of three statistical parameters-periodical mean, periodical standard deviation and skewness of the series. The results reveal that the periodical mean of the series generated by both Thomas–Fiering and ANN models are in good agreement with that of the historical series. However, periodical standard deviation and skewness coefficient of the series generated by Thomas–Fiering model is inferior to that of the series generated by ANN.

Analysis of the Velocity Tracking Control Problem for the 3D Evolutionary Navier--Stokes Equations

The velocity tracking problem for the evolutionary Navier--Stokes equations in three dimensions is studied. The controls are of distributed type and are submitted to bound constraints. The classical cost functional is modified so that a full analysis of the control problem is possible. First and second order necessary and sufficient optimality conditions are proved. A fully discrete scheme based on a discontinuous (in time) Galerkin approach, combined with conforming finite element subspaces in space, is proposed and analyzed. Provided that the time and space discretization parameters, $\tau$ and $h,$ respectively, satisfy $\tau \leq Ch^2$, the $L^2(\Omega_T)$ error estimates of order $O(h)$ are proved for the difference between the locally optimal controls and their discrete approximations. Finally, combining these techniques and the approach of Casas, Herzog, and Wachsmuth [SIAM J. Optim., 22 (2012), pp. 795--820], we extend our results to the case of $L^1(\Omega_T)$ type functionals that allow sparse c...

Numerical analysis of an adsorption dynamic model at the air–water interface

In this paper we deal with the numerical analysis of an adsorption dynamic model arising in a surfactant solution at the air–water interface; the diffusion model is considered together with the so-called Langmuir isotherm. An existence and uniqueness result is stated. Then, fully discrete approximations are introduced by using a finite element method and a hybrid combination of backward and forward Euler schemes. Error estimates are proved from which, under adequate additional regularity conditions, the linear convergence of the algorithm is derived assuming a dependence between both spatial and time discretization parameters. Finally, some numerical simulations are presented in order to demonstrate the accuracy of the algorithm and the behaviour of the solution for two commercially available surfactants.

Error estimates for the discretization of the velocity tracking problem

In this paper we are continuing our work (Casas and Chrysafinos, SIAM J Numer Anal 50(5):2281---2306, 2012), concerning a priori error estimates for the velocity tracking of two-dimensional evolutionary Navier---Stokes flows. The controls are of distributed type, and subject to point-wise control constraints. The discretization scheme of the state and adjoint equations is based on a discontinuous time-stepping scheme (in time) combined with conforming finite elements (in space) for the velocity and pressure. Provided that the time and space discretization parameters, $$\tau $$? and $$h$$h respectively, satisfy $$\tau \le Ch^2$$?≤Ch2, error estimates of order $$\mathcal {O}(h^2)$$O(h2) and $$\mathcal {O}(h^{\frac{3}{2}-\frac{2}{p}})$$O(h32-2p) with $$p > 3$$p>3 depending on the regularity of the target and the initial velocity, are proved for the difference between the locally optimal controls and their discrete approximations, when the controls are discretized by the variational discretization approach and by using piecewise-linear functions in space respectively. Both results are based on new duality arguments for the evolutionary Navier---Stokes equations.

On the existence of a solution for an adsorption dynamic model with the Langmuir isotherm

In this paper, we study an adsorption model arising in the dynamics of several surfactants at the air-water interface, where the Langmuir isotherm is employed for modelling the time-dependent surface concentration, providing a nonlinear dynamical boundary condition. Existence of a weak solution is proved by using the Rothe method for a semi-discrete problem in time. After obtaining some a priori estimates and passing to the limit in the time discretization parameter, we conclude that the original Langmuir problem has a bounded solution. An uniqueness result is also given.

A preconditioned MinRes solver for time‐periodic parabolic optimal control problems

SUMMARYThis work is devoted to the multiharmonic analysis of parabolic optimal control problems in a time‐periodic setting. In contrast to previous approaches, we include the cases of different control and observation domains, the observation in certain energy spaces, and the presence of control constraints. In all these cases, we propose a new preconditioned MinRes solver for the frequency domain equations and show that this solver is robust with respect to the space and time discretization parameters as well as the involved ‘bad’ model parameters of the state equation. Copyright © 2012 John Wiley & Sons, Ltd.

A Robust Preconditioned MinRes Solver for Distributed Time-Periodic Eddy Current Optimal Control Problems

This work is devoted to the construction and analysis of robust solvers for some distributed optimal control problems for time-periodic eddy current problems. We apply the multiharmonic approach to the optimality system and construct a new preconditioned MinRes solver for the system of frequency domain equations. We show that this solver is robust with respect to the space discretization and time discretization parameters as well as all involved “bad” parameters, like the conductivity, the reluctivity, and the regularization and cost parameters.

A Discontinuous Galerkin Time-Stepping Scheme for the Velocity Tracking Problem

The velocity tracking problem for the evolutionary Navier--Stokes equations in two dimensions is studied. The controls are of distributed type and are submitted to bound constraints. First and second order necessary and sufficient conditions are proved. A fully discrete scheme based on the discontinuous (in time) Galerkin approach, combined with conforming finite element subspaces in space, is proposed and analyzed. Provided that the time and space discretization parameters, $\tau$ and $h$, respectively, satisfy $\tau \leq Ch^2$, then $L^2$ error estimates of order $O(h)$ are proved for the difference between the locally optimal controls and their discrete approximations.

A linear stability analysis for nonlinear, grey, thermal radiative transfer problems

We present a new linear stability analysis of three time discretizations and Monte Carlo interpretations of the nonlinear, grey thermal radiative transfer (TRT) equations: the widely used “Implicit Monte Carlo” (IMC) equations, the Carter Forest (CF) equations, and the Ahrens–Larsen or “Semi-Analog Monte Carlo” (SMC) equations. Using a spatial Fourier analysis of the 1-D Implicit Monte Carlo (IMC) equations that are linearized about an equilibrium solution, we show that the IMC equations are unconditionally stable (undamped perturbations do not exist) if α, the IMC time-discretization parameter, satisfies 0.5<α⩽1. This is consistent with conventional wisdom. However, we also show that for sufficiently large time steps, unphysical damped oscillations can exist that correspond to the lowest-frequency Fourier modes. After numerically confirming this result, we develop a method to assess the stability of any time discretization of the 0-D, nonlinear, grey, thermal radiative transfer problem. Subsequent analyses of the CF and SMC methods then demonstrate that the CF method is unconditionally stable and monotonic, but the SMC method is conditionally stable and permits unphysical oscillatory solutions that can prevent it from reaching equilibrium. This stability theory provides new conditions on the time step to guarantee monotonicity of the IMC solution, although they are likely too conservative to be used in practice. Theoretical predictions are tested and confirmed with numerical experiments.

Bubble stabilized discontinuous Galerkin method for parabolic and elliptic problems

In this paper we give an analysis of a bubble stabilized discontinuous Galerkin method for elliptic and parabolic problems. The method consists of stabilizing the numerical scheme by enriching the discontinuous affine finite element space elementwise by quadratic bubbles. This approach leads to optimal convergence in the space and time discretization parameters.

Propagation of Gevrey regularity over long times for the fully discrete Lie Trotter splitting scheme applied to the linear Schrödinger equation

In this paper, we study the linear Schrodinger equation over the d-dimensional torus, with small values of the perturbing potential. We consider numerical approximations of the associated solutions obtained by a symplectic splitting method (to discretize the time variable) in combination with the Fast Fourier Transform algorithm (to discretize the space variable). In this fully discrete setting, we prove that the regularity of the initial datum is preserved over long times, i.e. times that are exponentially long with the time discretization parameter. We here refer to Gevrey regularity, and our estimates turn out to be uniform in the space discretization parameter. This paper extends (G. Dujardin and E. Faou, Numer. Math. 97 (2004) 493-535), where a similar result has been obtained in the semi-discrete situation, i.e. when the mere time variable is discretized and space is kept a continuous variable.

Galerkin Finite Element Methods with Symmetric Pressure Stabilization for the Transient Stokes Equations: Stability and Convergence Analysis

We consider the stability and convergence analysis of pressure stabilized finite element approximations of the transient Stokes equation. The analysis is valid for a class of symmetric pressure stabilization operators, but also for standard, inf-sup stable, velocity/pressure spaces without stabilization. Provided the initial data are chosen as a specific (method-dependent) Ritz-projection, we get unconditional stability and optimal convergence for both pressure and velocity approximations, in natural norms. For arbitrary interpolations of the initial data, a condition between the space and time discretization parameters has to be verified in order to guarantee pressure stability.

Uniformly exponentially stable approximations for a class of damped systems

We consider time semi-discrete approximations of a class of exponentially stable infinite-dimensional systems modeling, for instance, damped vibrations. It has recently been proved that for time semi-discrete systems, due to high frequency spurious components, the exponential decay property may be lost as the time step tends to zero. We prove that adding a suitable numerical viscosity term in the numerical scheme, one obtains approximations that are uniformly exponentially stable. This result is then combined with previous ones on space semi-discretizations to derive similar results on fully-discrete approximation schemes. Our method is mainly based on a decoupling argument of low and high frequencies, the low frequency observability property for time semi-discrete approximations of conservative linear systems and the dissipativity of the numerical viscosity on the high frequency components. Our methods also allow to deal directly with stabilization properties of fully discrete approximation schemes without numerical viscosity, under a suitable CFL type condition on the time and space discretization parameters.

A Priori Error Estimates for Space-Time Finite Element Discretization of Parabolic Optimal Control Problems Part I: Problems Without Control Constraints

In this paper we develop a priori error analysis for Galerkin finite element discretizations of optimal control problems governed by linear parabolic equations. The space discretization of the state variable is done using usual conforming finite elements, whereas the time discretization is based on discontinuous Galerkin methods. For different types of control discretizations we provide error estimates of optimal order with respect to both space and time discretization parameters. The paper is divided into two parts. In the first part we develop some stability and error estimates for space-time discretization of the state equation and provide error estimates for optimal control problems without control constraints. In the second part of the paper, the techniques and results of the first part are used to develop a priori error analysis for optimal control problems with pointwise inequality constraints on the control variable.

Numerical approximation of the boundary control for the wave equation with mixed finite elements in a square

This paper studies the numerical approximation of the boundary control for the wave equation in a square domain. It is known that the discrete and semi-discrete models obtained by discretizing the wave equation with the usual finite-difference or finite-element methods do not provide convergent sequences of approximations to the boundary control of the continuous wave equation as the mesh size goes to zero. Here, we introduce and analyse a new semi-discrete model based on the space discretization of the wave equation using a mixed finite-element method with two different basis functions for the position and velocity. The main theoretical result is a uniform observability inequality which allows us to construct a sequence of approximations converging to the minimal L2-norm control of the continuous wave equation. We also introduce a fully discrete system, obtained from our semi-discrete scheme, for which we conjecture that it provides a convergent sequence of discrete approximations as both h and Δt, the time discretization parameter, go to zero. We illustrate this fact with several numerical experiments.

First-order entropies for the Derrida-Lebowitz-Speer-Spohn equation

A logarithmic fourth-order parabolic equation in one space dimension with periodic boundary conditions is analyzed. Using a new semi-discrete approximation in time, a first-order entropy–entropy dissipation inequality is proved. Passing to the limit of vanishing time discretization parameter, some regularity results are deduced. Moreover, it is shown that the solution is strictly positive for large time if it does so initially.

Error estimates for a stabilized finite element method for the Oldroyd B model

In this paper, we study a new approximation scheme of transient viscoelastic fluid flow obeying an Oldroyd-B-type constitutive equation. The new stabilized formulation bases on the choice of a modified Euler method connected to the streamline upwinding Petrov–Galerkin (SUPG) method [M. Bensaada, D. Esselaoui, D. Sandri, Stabilization method for continuous approximation of transient convection problem, Numer. Methods Partial Differential Equations 21 (2004) 170–189], in order to stabilize the tensorial transport term of the Oldroyd derivative. Suppose that the continuous problem admits a sufficiently smooth and sufficiently small solution. A priori error estimates for the approximation in terms of the mesh parameter h and the time discretization parameter Δ t are derived.

Approximation of time-dependent, multi-component, viscoelastic fluid flow

In this article we analyse a fully discrete approximation to the time dependent viscoelasticity equations allowing for multicomponent fluid flow. The Oldroyd B constitutive equation is used to model the viscoelastic stress. For the discretization, time derivatives are replaced by backward difference quotients, and the non-linear terms are linearized by lagging appropriate factors. The modeling equations for the individual fluids are combined into a single system of equations using a continuum surface model. The numerical approximation is stabilized by using an SUPG approximation for the constitutive equation. Under a small data assumption on the true solution, existence of the approximate solution is proven. A priori error estimates for the approximation in terms of the mesh parameter h, the time discretization parameter Δ t, and the SUPG coefficient ν are also derived. Numerical simulations of viscoelastic fluid flow involving two immiscible fluids are also presented.

A Parallel High-accuracy Method for the First-order Evolution Equation in Hilbert and Banach Spaces

AbstractWe have developed a discretization method of an arbitrarily given order of accuracy with respect to the time discretization parameter for the first-order evolution differential equation in Banach space. The method includes two levels of parallelism: the operator exponential needed for the calculation of the evolution operator can be computed in parallel (inner parallelism) and then we can compute in parallel the evolution operator at various points of the time mesh (outer parallelism).