A streamline diffusion nonconforming finite element method for the time-dependent linearized Navier-Stokes equations

A nonconforming finite element method of finite difference streamline diffusion type is proposed to solve the time-dependent linearized Navier-Stokes equations. The backward Euler scheme is used for time discretization. Crouzeix-Raviart nonconforming finite element approximation, namely, nonconforming （P1）2 - P0 element, is used for the velocity and pressure fields with the streamline diffusion technique to cope with usual instabilities caused by the convection and time terms. Stability and error estimates are derived with suitable norms.

Global Attractors and Asymptotic Smoothing Effect for Navier-Stokes Equations with Linear Damping on R^2

This paper studies the long time behavior of solutions to the Navier-Stokes equations with linear damping on R^2. The authors prove the existence of L^2-global attractor and Hi-global attractor by showing that the corresponding semigroup is asymptotically compact. Thereafter, they establish that the two attractors are the same and thus reveal the asymptotic smoothing effect of the solutions.

Solution of Dirac Equation with the Time-Dependent Linear Potential in Non-Commutative Phase Space

In this paper, the time-dependent invariant of the Dirac equation with time-dependent linear potential has been constructed in non-commutative phase space. The corresponding analytical solution of the Dirac equation is presented by Lewis-Riesenfield invariant method.

Low-Storage Runge-Kutta Method for Simulating Time-Dependent Quantum Dynamics

A wide range of quantum systems are time-invariant and the corresponding dynamics is dic- tated by linear differential equations with constant coefficients. Although simple in math- ematical concept, the integration of these equations is usually complicated in practice for complex systems, where both the computational time and the memory storage become limit- ing factors. For this reason, low-storage Runge-Kutta methods become increasingly popular for the time integration. This work suggests a series of s-stage sth-order explicit Runge- Kutta methods specific for autonomous linear equations, which only requires two times of the memory storage for the state vector. We also introduce a 13-stage eighth-order scheme for autonomous linear equations, which has optimized stability region and is reduced to a fifth-order method for general equations. These methods exhibit significant performance improvements over the previous general-purpose low-stage schemes. As an example, we ap- ply the integrator to simulate the non-Markovian exciton dynamics in a 15-site linear chain consisting of perylene-bisimide derivatives.

Expansion of Linear Analysis of Rayleigh-Taylor Interface Instability of Metal Materials

The linear analysis of the Rayleigh-Taylor instability in metal material is extended from the perfect plastic constitutive model to the Johnson-Cook and Steinberg-Guinan constitutive model, and from the constant loading to a time-dependent loading. The analysis is applied to two Rayleigh-Taylor instability experiments in aluminum and vanadium with peak pressures of 20 GPa and 90 GPa, and strain rates of 6 × 106 s−1 and 3 × 107 s−1 respectively. When the time-dependent loading and the Steinberg-Guinan constitutive model are used in the linear analysis, the analytic results are in close agreement with experiments quantitatively, which indicates that the method in this paper is applicable to the Rayleigh-Taylor instability in aluminum and vanadium metal materials under high pressure and high strain rate. From these linear analyses, we find that the constitutive models and the loading process are of crucial importance in the linear analysis of the Rayleigh-Taylor instability in metal material, and a better understanding of the Rayleigh-Taylor instability in metals is gained. These results will serve as important references for evolving high-pressure, high-strain-rate experiments and numerical simulations.

The Effects of Rotation and Salt Concentration On Thermal Convection In a Linear Magneto-Fluid Layer Overlying a Porous Layer

A linear stability analysis is applied to a system consisting of a linear magneto-fluid layer overlying a porous layer affected by rotation and salt concentration on both layers. The flow in the fluid layer is governed by Navier-Stokes’s equations and while governed by Darcy-Brinkman’s law in the porous medium. Numerical solutions are obtained using Legendre polynomials. These solutions are studied through two modes of instability: stationary instability and overstability when the heat and the salt concentration are effected from above and below.

A NEW CRITERION FOR STABILITY OF LINEARTIME-DEPENDENT SYSTEM

In the paper, we establish a new criterion for stability of system dx/dt =A(t)x, which is based on the assumption that one leading principal submatrixof order r and its complementary submatrix in A(t) both have eigenvalues withonly negative real parts. And also a feasible method for decomposition andaggregation of large-scale systems is provided.

The Crouzeix-Raviart Type Nonconforming Finite Element Method for the Nonstationary Navier-Stokes Equations on Anisotropic Meshes

This paper is devoted to study the Crouzeix-Raviart （C-R） type nonconforming linear triangular finite element method （FEM） for the nonstationary Navier-Stokes equations on anisotropic meshes. By intro- ducing auxiliary finite element spaces, the error estimates for the velocity in the L2-norm and energy norm, as well as for the pressure in the L2-norm are derived.

Developing an Instantaneous Response Surface Method t-IRS for Time-Dependent Reliability Analysis

In practical engineering, many uncertain factors in loading or degradation of material properties may vary with time. Stochastic process modeling constitutes a suitable approach for describing these time-dependent uncertainties. By adopting this approach, however, the timedependent reliability calculation is a great challenge owing to the complexity and the huge computational burden. This paper presents a new instantaneous response surface method t-IRS for time-dependent reliability analysis. Different from the adaptive extreme response surface approach, the proposed method does not need to build and update surrogate models separately at each time node. It first uses the expansion optimal linear estimation method to discretize the stochastic processes into a set of independent standard normal variables together with some deterministic functions of time. Time is then treated as an independent one-dimensional variable. Next, initial samples are generated by Latin hypercube sampling, and the corresponding response values are calculated and utilized to construct an instantaneous response surrogate model of the Kriging type. The active learning method is applied to update the Kriging surrogate model until satisfactory accuracy is achieved. Finally, the instantaneous response surrogate model is used to compute the time-dependent reliability via Monte Carlo simulation. Four case studies are utilized to demonstrate the effectiveness of the ^-IRS method for time-dependent reliability analysis.

Effect of the Time-Dependent Atom-Field Couplings on Entanglement

A model containing two two-level atoms and a single-mode cavity is considered, and the effect of the time-dependent atom-field couplings on entanglement between the two atoms is studied. The results indicate that both for one-photon processes and for two-photon processes, the disappearance of the initial entanglement is delayed due to the linear modulation of the atom-field coupling coefficients as compared to the constant coupling model. The delayed time of the disappearance of the initial entanglement for the two-photon processes is much longer than that for the one-photon processes in the case of adiabatic variation.

Unconditional Optimal Error Estimates for the Transient Navier-Stokes Equations with Damping

In this paper,the transient Navier-Stokes equations with damping are considered.Firstly,the semi-discrete scheme is discussed and optimal error estimates are derived.Secondly,a linearized backward Euler scheme is proposed.By the error split technique,the Stokes operator and the H^(-1)-norm estimate,unconditional optimal error estimates for the velocity in the norms L^(∞)(L^(2)) and L^(∞)(H^(1)),and the pressure in the norm L^(∞)(L^(2))are deduced.Finally,two numerical examples are provided to confirm the theoretical analysis.

One-dimensional nonlinear consolidation of soft clay with the non-Darcian flow

Based on the non-Darcian flow law described by exponent m and threshold gradient i 1 under a low hydraulic gradient and the classical nonlinear relationships e-lgσ′ and e-lgk v (Mesri and Rokhsar, 1974), the governing equation of 1D nonlinear consolidation was modified by considering both uniform distribution of self-weight stress and linear increment of self-weight stress. The numerical solutions for the governing equation were derived by the finite difference method (FDM). Moreover, the solutions were verified by comparing the numerical results with those by analytical method under a specific case. Finally, consolidation behavior under different parameters was investigated, and the results show that the rate of 1D nonlinear consolidation will slow down when the non-Darcian flow law is considered. The consolidation rate with linear increment of self-weight stress is faster than that with uniform distribution one. Compared to Darcy's flow law, the influence of parameters describing non-linearity of soft soil on consolidation behavior with non-Darcian flow has no significant change.

Wavelet estimation of the diffusion coefficient in time dependent diffusion models

The estimation problem for diffusion coefficients in diffusion processes has been studied in many papers,where the diffusion coefficient function is assumed to be a 1-dimensional bounded Lipschitzian function of the state or the time only.There is no previous work for the nonparametric estimation of time-dependent diffusion models where the diffusion coefficient depends on both the state and the time.This paper introduces and studies a wavelet estimation of the time-dependent diffusion coefficient under a more general assumption that the diffusion coefficient is a linear growth Lipschitz function.Using the properties of martingale,we translate the problems in diffusion into the nonparametric regression setting and give the Lr convergence rate.A strong consistency of the estimate is established.With this result one can estimate the time-dependent diffusion coefficient using the same structure of the wavelet estimators under any equivalent probability measure.For example,in finance,the wavelet estimator is strongly consistent under the market probability measure as well as the risk neutral probability measure.

Semi-Implicit Interior Penalty Discontinuous Galerkin Methods for Viscous Compressible Flows

We deal with the numerical solution of the Navier-Stokes equations describing a motion of viscous compressible fluids.In order to obtain a sufficiently stable higher order scheme with respect to the time and space coordinates,we develop a combination of the discontinuous Galerkin finite element(DGFE)method for the space discretization and the backward difference formulae(BDF)for the time discretization.Since the resulting discrete problem leads to a system of nonlinear algebraic equations at each time step,we employ suitable linearizations of inviscid as well as viscous fluxes which give a linear algebraic problem at each time step.Finally,the resulting BDF-DGFE scheme is applied to steady as well as unsteady flows and achieved results are compared with reference data.

DECOUPLED, ENERGY STABLE SCHEME FOR HYDRODYNAMIC ALLEN-CAHN PHASE FIELD MOVING CONTACT LINE MODEL

In this paper, we present an efficient energy stable scheme to solve a phase field model incorporating contact line condition. Instead of the usually used Cahn-Hilliard type phase equation, we adopt the Allen-Cahn type phase field model with the static contact line boundary condition that coupled with incompressible Navier-Stokes equations with Navier boundary condition. The projection method is used to deal with the Navier-Stokes equa- tions and an auxiliary function is introduced for the non-convex Ginzburg-Landau bulk potential. We show that the scheme is linear, decoupled and energy stable. Moreover, we prove that fully discrete scheme is also energy stable. An efficient finite element spatial discretization method is implemented to verify the accuracy and efficiency of proposed schemes. Numerical results show that the proposed scheme is very efficient and accurate.