High-order approximations for an incompressible viscous flow on a rough boundary

In this paper, we propose approximations of fluid flow that could be used for obtaining wall laws of higher order. We consider the two-dimensional laminar fluid flow, modeled by the incompressible Stokes system in a straight channel with a rough side. The roughness is periodic and the ratio of the amplitude of the rough part and the size of the flow domain is denoted by ?, being a small number. We impose periodic boundary conditions on the flow. We generalize the boundary layers needed for the construction of flow approximations of higher order with respect to ?. The existence of the layers and their features are discussed. Finally we give the error estimates for the approximations and establish an explicit wall law.     

Numerical simulation of axisymmetric afterbody flows with jet exhaust

A fifth-order accurate compact difference scheme was used to compute the flow over an axisymmetric afterbody with jet exhaust. The solution was based on the mass-averaged Navier-Stokes equations combined with a two-parameter differential model of turbulence. The computations were performed on a specially generated mesh such that the flow in the exterior and interior of the nozzle could be described simultaneously. Numerical results are presented for various external flow conditions and various chamber pressures.     

Implicit finite element schemes for stationary compressible particle-laden gas flows

The derivation of macroscopic models for particle-laden gas flows is reviewed. Semi-implicit and Newton-like finite element methods are developed for the stationary two-fluid model governing compressible particle-laden gas flows. The Galerkin discretization of the inviscid fluxes is potentially oscillatory and unstable. To suppress numerical oscillations, the spatial discretization is performed by a high-resolution finite element scheme based on algebraic flux correction. A multidimensional limiter of TVD type is employed. An important goal is the efficient computation of stationary solutions in a wide range of Mach numbers. This is a challenging task due to oscillatory correction factors associated with TVD-type flux limiters and the additional strong nonlinearity caused by interfacial coupling terms. A semi-implicit scheme is derived by a time-lagged linearization of the nonlinear residual, and a Newton-like method is obtained in the limit of infinite CFL numbers. The original Jacobian is replaced by a low-order approximation. Special emphasis is laid on the numerical treatment of weakly imposed boundary conditions. It is shown that the proposed approach offers unconditional stability and faster convergence rates for increasing CFL numbers. The strongly coupled solver is compared to operator splitting techniques, which are shown to be less robust.     

求解一维定常对流扩散反应方程的混合型紧致差分格式

借助显式紧致格式和隐式紧致格式的思想,基于截断误差余项修正,并结合原方程本身,构造出了一种求解一维定常对流扩散反应方程的高精度混合型紧致差分格式.格式仅用到三个点上的未知函数值及一阶导数值,而一阶导数值利用四阶Pade格式进行计算,格式整体具有四阶精度.数值实验结果验证了格式的精确性和可靠性.     

四维抛物型方程的高精度分支稳定的显式差分格式

对四维抛物型方程构造了一个高精度显格式,格式的稳定性条件为r=Δt/Δx2=△t/Δy2=△t/△z2=Δt/Δw2<1/2,截断误差阶达到O(Δt2 Δx4),通过数值实验,表明本文理论分析的正确性和文中格式较同类格式的优越性.     

Finite element simulation of compressible particle-laden gas flows

A macroscopic two-fluid model of compressible particle-laden gas flows is considered. The governing equations are discretized by a high-resolution finite element method based on algebraic flux correction. A multidimensional limiter of TVD type is employed to constrain the local characteristic variables for the continuous gas phase and conservative fluxes for a suspension of solid particles. Special emphasis is laid on the efficient computation of steady state solutions at arbitrary Mach numbers. To avoid stability restrictions and convergence problems, the characteristic boundary conditions are imposed weakly and treated in a fully implicit manner. A two-way coupling via the interphase drag force is implemented using operator splitting. The Douglas-Rachford scheme is found to provide a robust treatment of the interphase exchange terms within the framework of a fractional-step solution strategy. Two-dimensional simulation results are presented for a moving shock wave and for a steady nozzle flow.     

Ergodic measures of geodesic flows on compact Lie groups

Let Ψ be the geodesic flow associated with a two-sided invariant metric on a compact Lie group. In this paper, we prove that every ergodic measure μ of Ψ is supported on the unit tangent bundle of a flat torus. As an application, all Lyapunov exponents of μ are zero hence μ is not hyperbolic. Our underlying manifolds have nonnegative curvature (possibly strictly positive on some sections), whereas in contrast, all geodesic flows related to negative curvature are Anosov hence every ergodic measure is hyperbolic.     

High‐order compact schemes for fractional differential equations with mixed derivatives

In this article, we consider two‐dimensional fractional subdiffusion equations with mixed derivatives. A high‐order compact scheme is proposed to solve the problem. We establish a sufficient condition and show that the scheme converges with fourth order in space and second order in time under this condition.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2141–2158, 2017     

A compact ADI scheme for the two dimensional time fractional diffusion‐wave equation in polar coordinates

In this article, we consider finite difference schemes for two dimensional time fractional diffusion‐wave equations on an annular domain. The problem is formulated in polar coordinates and, therefore, has variable coefficients. A compact alternating direction implicit scheme with accuracy order is derived, where τ, h₁, h₂ are the temporal and spatial step sizes, respectively. The stability and convergence of the proposed scheme are studied using its matrix form by the energy method. Numerical experiments are presented to support the theoretical results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1692–1712, 2015     

Fourth-order alternating direction implicit compact finite difference schemes for two-dimensional Schrödinger equations

In this paper, alternating direction implicit compact finite difference schemes are devised for the numerical solution of two-dimensional Schrödinger equations. The convergence rates of the present schemes are of order O(h⁴+τ²). Numerical experiments show that these schemes preserve the conservation laws of charge and energy and achieve the expected convergence rates. Representative simulations show that the proposed schemes are applicable to problems of engineering interest and competitive when compared to other existing procedures.     

Conservative compact difference schemes for the coupled nonlinear schrödinger system

In this article, some conservative compact difference schemes are explored for the strongly coupled nonlinear schrödinger system. After transforming the scheme into matrix form, we prove the existence and uniqueness, convergence and stability of the difference solutions for one nonlinear scheme in the norm by using some techniques of matrix theory. Numerical results show that one nonlinear scheme is the most efficient of all the compact schemes constructed here. It allows much larger time steps than the others. The second most efficient compact scheme is a linear one. We then give numerical simulations to two soliton interactions for the two most efficient compact schemes. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 749–772, 2014     

On the structure of internal dissipation of composite compact schemes for gasdynamic simulation

The structure of the internal dissipation terms in composite compact schemes intended for gasdynamic simulation is considered. The main cause of the insufficient stability of high-order accurate schemes is indicated. A method for controlling the dissipative properties of schemes is proposed that makes it possible to compute compressible gas flows with strong shock waves. The supersonic turbulent unsteady flow past a two-dimensional cavity directed toward the stream is computed.     

一维非线性Schrödinger 方程的两个无条件收敛的守恒紧致差分格式

本文对一维非线性 Schrödinger 方程给出两个紧致差分格式, 运用能量方法和两个新的分析技 巧证明格式关于离散质量和离散能量守恒, 而且在最大模意义下无条件收敛. 对非线性紧格式构造了 一个新的迭代算法, 证明了算法的收敛性, 并在此基础上给出一个新的线性化紧格式. 数值算例验证 了理论分析的正确性, 并通过外推进一步提高了数值解的精度.     

A large deviations principle for stochastic flows of viscous fluids

We study the well-posedness of a stochastic differential equation on the two dimensional torus ${\mathbb{T}}^2$, driven by an infinite dimensional Wiener process with drift in the Sobolev space $L^2(0,T;H^1\left({\mathbb{T}}^2\right))$. The solution corresponds to a stochastic Lagrangian flow in the sense of DiPerna Lions. By taking into account that the motion of a viscous incompressible fluid on the torus can be described through a suitable stochastic differential equation of the previous type, we study the inviscid limit. By establishing a large deviations principle, we show that, as the viscosity goes to zero, the Lagrangian stochastic Navier–Stokes flow approaches the Euler deterministic Lagrangian flow with an exponential rate function.     

Fourth‐order compact difference schemes for 1D nonlinear Kuramoto–Tsuzuki equation

In this article, first, we establish some compact finite difference schemes of fourth‐order for 1D nonlinear Kuramoto–Tsuzuki equation with Neumann boundary conditions in two boundary points. Then, we provide numerical analysis for one nonlinear compact scheme by transforming the nonlinear compact scheme into matrix form. And using some novel techniques on the specific matrix emerged in this kind of boundary conditions, we obtain the priori estimates and prove the convergence in norm. Next, we analyze the convergence and stability for one of the linearized compact schemes. To obtain the maximum estimate of the numerical solutions of the linearized compact scheme, we use the mathematical induction method. The treatment is that the convergence in norm is obtained as well as the maximum estimate, further the convergence in norm. Finally, numerical experiments demonstrate the theoretical results and show that one of the linearized compact schemes is more accurate, efficient and robust than the others and the previous. It is worthwhile that the compact difference methods presented here can be extended to 2D case. As an example, we present one nonlinear compact scheme for 2D Ginzburg–Landau equation and numerical tests show that the method is accurate and effective. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 2080–2109, 2015     

Direct numerical simulation of turbulent flows in eccentric pipes

A numerical algorithm was developed for solving the incompressible Navier-Stokes equations in curvilinear orthogonal coordinates. The algorithm is based on a central-difference discretization in space and on a third-order accurate semi-implicit Runge-Kutta scheme for time integration. The discrete equations inherit some properties of the original differential equations, in particular, the neutrality of the convective terms and the pressure gradient in the kinetic energy production. The method was applied to the direct numerical simulation of turbulent flows between two eccentric cylinders. Numerical computations were performed at Re = 4000 (where the Reynolds number Re was defined in terms of the mean velocity and the hydraulic diameter). It was found that two types of flow develop depending on the geometric parameters. In the flow of one type, turbulent fluctuations were observed over the entire cross section of the pipe, including the narrowest gap, where the local Reynolds number was only about 500. The flow of the other type was divided into turbulent and laminar regions (in the wide and narrow parts of the gap, respectively).     

The Neumann stability of high-order symmetric schemes for convection-diffusion problems

We find a series of sufficient conditions for the Neumann stability of the leapfrog-Euler scheme of arbitrary even order in the spatial coordinates for the three-dimensional convection-diffusion equation.