Numerical analysis of a corrected Smagorinsky model

The classical Smagorinsky model's solution is an approximation to a (resolved) mean velocity. Since it is an eddy viscosity model, it cannot represent a flow of energy from unresolved fluctuations to the (resolved) mean velocity. This model has recently been corrected to incorporate this flow and still be well-posed. Herein we first develop some basic properties of the corrected model. Next, we perform a complete numerical analysis of two algorithms for its approximation. They are tested and proven to be effective.     

Numerical approximation of a quantum drift-diffusion model

This Note is devoted to the discretization and numerical simulation of a new quantum drift-diffusion model that was recently derived. We define an implicit numerical scheme which is equivalent to a convex minimization problem and which preserves the physical properties of the continuous model: charge conservation, positivity of the density and dissipation of an entropy. We illustrate these results by some numerical simulations. To cite this article: S. Gallego, F. Méhats, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

A two-level additive Schwarz method for the Morley nonconforming element approximation of a nonlinear biharmonic equation

In this paper, we consider the well known Morley nonconformingelement approximation of a nonlinear biharmonic equation whichis related to the well-known two-dimensional Navier–Stokesequations. Firstly, optimal energy and H¹-norm estimates areobtained. Secondly, a two-level additive Schwarz method is presentedfor the discrete nonlinear algebraic system. It is shown thatif the Reynolds number is sufficiently small, the two-levelSchwarz method is optimal, i.e. the convergence rate of theSchwarz method is independent of the mesh size and the numberof subdomains.     

Numerical approximation of the fractional Cahn-Hilliard equation by operator splitting method

Numerical Algorithms - In this paper, we consider a fast explicit operator splitting method for a fractional Cahn-Hilliard equation with spatial derivative $(-{varDelta })^{frac {alpha }{2}}$...     

Numerical approximation for a Baer-Nunziato model of two-phase flows

We present a well-balanced numerical scheme for approximating the solution of the Baer-Nunziato model of two-phase flows by balancing the source terms and discretizing the compaction dynamics equation. First, the system is transformed into a new one of three subsystems: the first subsystem consists of the balance laws in the gas phase, the second subsystem consists of the conservation law of the mass in the solid phase and the conservation law of the momentum of the mixture, and the compaction dynamic equation is considered as the third subsystem. In the first subsystem, stationary waves are used to build up a well-balanced scheme which can capture equilibrium states. The second subsystem is of conservative form and thus can be numerically treated in a standard way. For the third subsystem, the fact that the solid velocity is constant across the solid contact suggests us to compose the technique of the Engquist-Osher scheme. We show that our scheme is capable of capturing exactly equilibrium states. Moreover, numerical tests show the convergence of approximate solutions to the exact solution.     

Numerical approximation of a thermoviscoelastic problem to the contact of two rods by a penalty method

In this article we analyze a numerical approximation by the finite element method of a system modeling the thermoviscoelastic contact of two rods. The numerical scheme is based on an auxiliary penalized problem. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004     

Energy dissipation in the Smagorinsky model of turbulence

The Smagorinsky model often severely over-dissipates flows and, consistently, previous estimates of its energy dissipation rate blow up as $Re\to \infty$. This report estimates time averaged model dissipation, $〈{\epsilon }_S〉$, under periodic boundary conditions as$〈{\epsilon }_S〉\le 2\frac{U^3}{L}+{\mathrm{Re}}^{-1}\frac{U^3}{L}+\frac{32}{27}{C}_S^2{\left(\frac{\delta }{L}\right)}^2\frac{U^3}{L}\text{,}$ where $U,L$ are global velocity and length scales and $C_S\simeq 0.1,\delta <1$ are model parameters. Thus, in the absence of boundary layers, the Smagorinsky model does not over dissipate.     

Numerical approximation of the Oldroyd‐B model by the weighted least‐squares/discontinuous Galerkin method

We consider a numerical method for the Oldroyd‐B model of viscoelastic fluid flows by a combination of the weighted least‐squares (WLS) method and the discontinuous Galerkin (DG) finite element method. The constitutive equation is decoupled from the momentum and continuity equations, and the approximate solution is computed iteratively by solving the Stokes problem and a linearized constitutive equation using WLS and DG, respectively. An a priori error estimate for the WLS/DG method is derived and numerical results supporting the estimate are presented. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

Maximum approximation of the best guaranteed result in a two-level hierarchical game

We consider the well-known two-level hierarchical game between the center and the production divisions in the presence of uncontrolled external factors. The problem of the best guaranteed result for the center in this game is approximated by a maxmin problem with separable variables. A stochastic quasigradient solution method is considered.Translated from Programmno-apparatnye Sredstva i Matematicheskoe Obespechenie Vychislitel'nykh Sistem, pp. 136–142, 1994.     

Numerical experiments with a method of successive approximation for conformal mapping

A method of successive approximation for the conformal mapping of simply connected regions onto the unit disc and numerical experiments with it are described. The experiments gave unexpectedly good results.

---

Zusammenfassung Es werden ein Schmiegungsverfahren für die konforme Abbildung einfach zusammenhängender Gebiete auf den Einheitskreis und numerische Experimente damit beschrieben. Letztere ergaben unerwartet gute Resultate.

---

---

Born December 6, 1942; killed August 31, 1979, while climbing in the mountains of Canada.     

Numerical approximation of the conservative Allen–Cahn equation by operator splitting method

In this paper, a second‐order fast explicit operator splitting method is proposed to solve the mass‐conserving Allen–Cahn equation with a space–time‐dependent Lagrange multiplier. The space–time‐dependent Lagrange multiplier can preserve the volume of the system and keep small features. Moreover, we analyze the discrete maximum principle and the convergence rate of the fast explicit operator splitting method. The proposed numerical scheme is of spectral accuracy in space and of second‐order accuracy in time, which greatly improves the computational efficiency. Numerical experiments are presented to confirm the accuracy, efficiency, mass conservation, and stability of the proposed method. Copyright © 2017 John Wiley & Sons, Ltd.     

Model reduction of semiaffinely parameterized partial differential equations by two-level affine approximation

We propose an improvement to the reduced basis method for parametric partial differential equations. An assumption of affine parameterization leads to an efficient offline–online decomposition when the problem is solved for many different parametric configurations. We consider an advection–diffusion problem, where the diffusive term is non-affinely parameterized and treated with a two-level affine approximation given by the empirical interpolation method. The offline stage and a posteriori error estimation is performed using the coarse-level approximation, while the fine-level approximation is used to perform a correction iteration that reduces the actual error of the reduced basis approximation while keeping the same certified error bounds.     

Numerical approximation of highly oscillatory integrals on semi-finite intervals by steepest descent method

A numerical steepest descent method, based on the Laguerre quadrature rule, is developed for integration of one-dimensional highly oscillatory functions on [0,?∞?) of a general class. It is shown that if the integrand is analytic, then in the absence of stationary points, the method is rapidly convergent. The method is extended to the case when there are a finite number of stationary points in [0,?∞?). It can be further extended to the case when the integrand is only smooth to some degree (not necessarily analytic). We illustrate the theoretical results using some numerical experiences.     

Numerical approximation of a metastable system

Using the Becker-Döring cluster equations as an example,we highlight some of the problems that can arise in the numericalapproximation of dynamical systems with slowly varying solutions.We describe the Becker-Döring model, summarize some ofits properties and construct a numerical approximation whichallows accurate and efficient computation of solutions in thelong, slowly varying metastable phase. We use the approximationto obtain test results and discuss the clear relationship betweenthem and equilibrium solutions of the Becker-Döring equations.     

Phase transition in a model two-level system

A two-level Bose-system interacting with a multimode Fermi-field is considered. The path integration method is used to obtain the spectrum of the collective excitations below the phase transition point in the superradiant state. The question of phase transition stability is investigated.     

Numerical approximation of a stochastic age-structured population model in a polluted environment with Markovian switching

In this paper, a stochastic age-structured population model with Markovian switching is investigated in a polluted environment. Both the stochastic disturbance of environment and the Markovian switching are incorporated into the model. By Itô formula and several assumptions, the boundedness in the qth moment of exact solutions of model are proved. Furthermore, making use of truncated Euler–Maruyama (EM) method, the strong convergence criterion of numerical approximation in the qth moment is established, and the rate of convergence is estimated. Numerical simulations are carried out to illustrate the theoretical results. Our results indicate that the truncated EM method can be used for stochastic age-structured population system in a polluted environment.     

Numerical analysis of reaction front propagation model under Boussinesq approximation

This paper deals with numerical analysis of system coupling Navier–Stokes equations with two non‐linear reaction–diffusion equations. This system modelize a propagation of reaction front at the case of polymerization. A finite elements approximation is presented, the existence and uniqueness are established. Optimal error estimates are given. Copyright © 2003 John Wiley & Sons, Ltd.