Homogenization of transport equations: A simple PDE approach to the Kubo formula

We are interested in the behavior with respect to the small parameter ?>0 of solutions ρ? of the conservative transport(-diffusion) equation t_∂ρ?+∇x(ρ?u?)=ηΔxρ?, with η?0, driven by a large random velocity field: |u?|=O(1/?). Assuming that the velocity does not have long-time memory we justify the convergence of the expectation Eρ? to the solution of a diffusion equation. This question has been widely investigated; here we present a simple proof which only relies on PDE tools.     

On Tsertsvadze's difference scheme for the Kuramoto-Tsuzuki equation

In this paper, we have proved the second-order convergence in L_∞ norm of the Tsertsvadze's difference scheme for the Kuramoto-Tsuzuki equation. The existence, uniqueness and iterative algorithm are also discussed in detail. Furthermore, a L_∞ second order convergent linearized difference scheme is given for inhomogeneous equation. All results are obtained without any restrictions on the meshsizes. At last a numerical example is presented.     

A linearized difference scheme for semilinear parabolic equations with nonlinear absorbing boundary conditions

A novel three level linearized difference scheme is proposed for the semilinear parabolic equation with nonlinear absorbing boundary conditions. The solution of this problem will blow up in finite time. Hence this difference scheme is coupled with an adaptive time step size, i.e., when the solution tends to infinity, the time step size will be smaller and smaller. Furthermore, the solvability, stability and convergence of the difference scheme are proved by the energy method. Numerical experiments are also given to demonstrate the theoretical second order convergence both in time and in space in L_∞-norm.     

Blowing up of a finite difference solution tou t = uxx + u2

This paper studies a finite difference approximation to the similinear heat equation (1) with special emphasis on the case when the exact solution blows up with the blowing-up timeT _∞. The key results will be given in Propositions 1 and 2. Proposition 1 states the local convergence, i.e., the convergence of the proposed finite difference solution to the exact solution in any fixed time interval 0 ?t ? T, whereT < T _∞. Proposition 2 states the convergence of the numerical blowing-up time to the exact oneT _∞.     

A two-grid method for elliptic problem with boundary layers

A two-grid method for the elliptic equation with a small parameter ε multiplying the highest derivative is investigated. The difference schemes with the property of ε-uniform convergence on a uniform mesh and on Shishkin mesh are considered. In both cases, a two-grid method for resolving the difference scheme is investigated. A two-grid method has features that are concerned with a uniform convergence of a difference scheme. To increase the accuracy, the Richardson extrapolation in two-grid method is applied. Numerical results are discussed.     

A method of asymptotic constructions with improved accuracy for a quasilinear singularly perturbed parabolic convection-diffusion equation

The Dirichlet problem on an interval for quasilinear singularly perturbed parabolic convection-diffusion equation is considered. The higher order derivative of the equation is multiplied by a parameter ε that takes any values from the half-open interval (0, 1]. For this type of linear problems, the order of the ε-uniform convergence (with respect to x and t) for the well-known schemes is not higher than unity (in the maximum norm). For the boundary value problem under consideration, grid approximations are constructed that converge ε-uniformly at the rate of O(N ^?2ln² N + N ^?2 ₀), where N + 1 and N ₀ + 1 are the numbers of the mesh points with respect to x and t, respectively. On the x axis, piecewise uniform meshes that condense in the boundary layer are used. If the parameter value is small compared to the effective step of the spatial grid, the domain decomposition method is used, which is motivated by “asymptotic constructions.” Monotone approximations of “auxiliary” subproblems describing the main terms of the asymptotic expansion of the solution outside a neighborhood of the boundary layer neighborhood are used. In the neighborhood of the boundary layer (of the width O(ε ln N)) the first derivative with respect to x is approximated by the central difference derivative. These subproblems are successively solved in the subdomains on uniform grids. If the parameter values are not sufficiently small (compared to the effective step of the mesh with respect to x), the classical implicit difference schemes approximating the first derivative with respect to x by the central difference derivative are applied. To improve the accuracy in t, the defect correction technique is used. Notice that the calculation of the solution of the constructed difference scheme (the scheme based on the method of asymptotic constructions) can be considerably simplified for sufficiently small values of the parameter ε.     

Order h4 difference methods for a class of singular two space elliptic boundary value problems

In this article, we derive difference methods of O(h⁴) for solving the system of two space nonlinear elliptic partial differential equations with variable coefficients having mixed derivatives on a uniform square grid using nine grid points. We obtain two sets of fouth-order difference methods; one in the absence of mixed derivatives, second when the coefficients of u_xy are not equal to zero and the coefficients of u_xx and u_yy are equal. There do not exist fourth-order schemes involving nine grid points for the general case. The method having two variables has been tested on two-dimensional viscous, incompressible steady-state Navier-Stokes' model equations in polar coordinates. The proposed difference method for scalar equation is also applied to the Poisson's equation in polar coordinates. Some numerical examples are provided to illustrate the fourth-order convergence of the proposed methods.     

Uniform grid approximation of nonsmooth solutions to the mixed boundary value problem for a singularly perturbed reaction-diffusion equation in a rectangle

A mixed boundary value problem for a singularly perturbed reaction-diffusion equation in a square is considered. A Neumann condition is specified on one side of the square, and a Dirichlet condition is set on the other three. It is assumed that the coefficient of the equation, its right-hand side, and the boundary values of the desired solution or its normal derivative on the sides of the square are smooth enough to ensure the required smoothness of the solution in a closed domain outside the neighborhoods of the corner points. No compatibility conditions are assumed to hold at the corner points. Under these assumptions, the desired solution in the entire closed domain is of limited smoothness: it belongs only to the Hölder class C ^μ, where μ ∈ (0, 1) is arbitrary. In the domain, a nonuniform rectangular mesh is introduced that is refined in the boundary domain and depends on a small parameter. The numerical solution to the problem is based on the classical five-point approximation of the equation and a four-point approximation of the Neumann boundary condition. A mesh refinement rule is described under which the approximate solution converges to the exact one uniformly with respect to the small parameter in the L _∞ ^h norm. The convergence rate is O(N ^?2ln² N), where N is the number of mesh nodes in each coordinate direction. The parameter-uniform convergence of difference schemes for mixed problems without compatibility conditions at corner points was not previously analyzed.     

Convergence and stability of a stabilized finite volume method for the stationary Navier-Stokes equations

In this paper, a new stabilized finite volume method is studied and developed for the stationary Navier-Stokes equations. This method is based on a local Gauss integration technique and uses the lowest equal order finite element pair P ₁–P ₁ (linear functions). Stability and convergence of the optimal order in the H ¹-norm for velocity and the L ²-norm for pressure are obtained. A new duality for the Navier-Stokes equations is introduced to establish the convergence of the optimal order in the L ²-norm for velocity. Moreover, superconvergence between the conforming mixed finite element solution and the finite volume solution using the same finite element pair is derived. Numerical results are shown to support the developed convergence theory.     

Stability and convergence of fully discrete finite element schemes for the acoustic wave equation

In this paper, we investigate the stability and convergence of some fully discrete finite element schemes for solving the acoustic wave equation where a discontinuous Galerkin discretization in space is used. We first review and compare conventional time-stepping methods for solving the acoustic wave equation. We identify their main properties and investigate their relationship. The study includes the Newmark algorithm which has been used extensively in applications. We present a rigorous stability analysis based on the energy method and derive sharp stability results covering some well-known CFL conditions. A convergence analysis is carried out and optimal a priori error estimates are obtained. For sufficiently smooth solutions, we demonstrate that the maximal error in the L ²-norm error over a finite time interval converges optimally as O(h ^p+1+??t ^s ), where p denotes the polynomial degree, s=1 or 2, h the mesh size, and ??t the time step.     

Value distribution of meromorphic solutions of certain difference Painlevé equations

The Borel exceptional value and the exponents of convergence of poles, zeros and fixed points of finite order transcendental meromorphic solutions for difference Painlevé I and II equations are estimated. And the forms of rational solutions of the difference Painlevé II equation and the autonomous difference Painlevé I equation are also given. It is also proved that the non-autonomous difference Painlevé I equation has no rational solution.     

Long-time behavior of the difference solutions to the 2D GKS equation

We study the long-time behavior of the finite difference solution to the generalized Kuramoto-Sivashinsky equation in two space dimensions with periodic boundary conditions. The unique solvability of numerical solution is shown. It is proved that there exists a global attractor of the discrete dynamical system and the upper semicontinuity d(A_h,τ,A)→0. Finally, we obtain the long-time stability and convergence of the difference scheme. Our results show that the difference scheme can effectively simulate the infinite dimensional dynamical systems.     

A three level linearized compact difference scheme for the Cahn-Hilliard equation

This article is devoted to the study of high order accuracy difference methods for the Cahn-Hilliard equation.A three level linearized compact difference scheme is derived.The unique solvability and unconditional convergence of the difference solution are proved.The convergence order is O(τ 2 + h 4 ) in the maximum norm.The mass conservation and the non-increase of the total energy are also verified.Some numerical examples are given to demonstrate the theoretical results.     

KP<Subscript>1</Subscript> acceleration scheme for inner iterations consistent with the weighted diamond differencing scheme for the transport equation in three-dimensional geometry

For the transport equation in three-dimensional (r, ?, z) geometry, a KP₁ acceleration scheme for inner iterations that is consistent with the weighted diamond differencing (WDD) scheme is constructed. The P ₁ system for accelerating corrections is solved by an algorithm based on the cyclic splitting method (SM) combined with Gaussian elimination as applied to auxiliary systems of two-point equations. No constraints are imposed on the choice of the weights in the WDD scheme, and the algorithm can be used, for example, in combination with an adaptive WDD scheme. For problems with periodic boundary conditions, the two-point systems of equations are solved by the cyclic through-computations method elimination. The influence exerted by the cycle step choice and the convergence criterion for SM iterations on the efficiency of the algorithm is analyzed. The algorithm is modified to threedimensional (x, y, z) geometry. Numerical examples are presented featuring the KP₁ scheme as applied to typical radiation transport problems in three-dimensional geometry, including those with an important role of scattering anisotropy. A reduction in the efficiency of the consistent KP₁ scheme in highly heterogeneous problems with dominant scattering in non-one-dimensional geometry is discussed. An approach is proposed for coping with this difficulty. It is based on improving the monotonicity of the difference scheme used to approximate the transport equation.     

Central limit theorem for a class of one-dimensional kinetic equations

We introduce a class of kinetic-type equations on the real line, which constitute extensions of the classical Kac caricature. The collisional gain operators are defined by smoothing transformations with rather general properties. By establishing a connection to the central limit problem, we are able to prove long-time convergence of the equation??s solutions toward a limit distribution. For example, we prove that if the initial condition belongs to the domain of normal attraction of a certain stable law ?? _??, then the limit is a scale mixture of ?? _??. Under some additional assumptions, explicit exponential rates for the convergence to equilibrium in Wasserstein metrics are calculated, and strong convergence of the probability densities is shown.     

Lacunary systems and generalized linear processes

Herein we obtain several almost sure and L_p convergence properties of generalized linear processes. When the generalized linear process is generated by an orthonormal S_p system (which includes the classical i.i.d. white noise, L_p bounded martingale difference sequences and weakly multiplicative sequences as special cases), it is shown that these convergence properties are related to certain second order properties of the process.     

Implicit compact difference schemes for the fractional cable equation

In this paper, we propose two implicit compact difference schemes for the fractional cable equation. The first scheme is proved to be stable and convergent in l_∞-norm with the convergence order O(τ + h⁴) by the energy method, where new inner products defined in this paper gives great convenience for the theoretical analysis. Numerical experiments are presented to demonstrate the accuracy and effectiveness of the two compact schemes. The computational results show that the two new schemes proposed in this paper are more accurate and effective than the previous.     

Finite difference on grids with nearly uniform cell area and line spacing for the wave equation on complex domains

A finite difference time-dependent numerical method for the wave equation, supported by recently derived novel elliptic grids, is analyzed. The method is successfully applied to single and multiple two-dimensional acoustic scattering problems including soft and hard obstacles with complexly shaped boundaries. The new grids have nearly uniform cell area (J-grids) and nearly uniform grid line spacing (αγ-grids). Numerical experiments reveal the positive impact of these two grid properties on the scattered field convergence to its harmonic steady state. The restriction imposed by stability conditions on the time step size is relaxed due to the near uniformity cell areas and grid line spacing. As a consequence, moderately large time steps can be used for relatively fine spatial grids resulting in greater accuracy at a lower computational cost. Also, numerical solutions for wave problems inside annular regions of complex shapes are obtained. The use of the new grids results in late time stability in contrast with other classical finite difference time-dependent methods.     

Computer difference scheme for a singularly perturbed elliptic convection–diffusion equation in the presence of perturbations

A grid approximation of a boundary value problem for a singularly perturbed elliptic convection–diffusion equation with a perturbation parameter ε, ε ∈ (0,1], multiplying the highest order derivatives is considered on a rectangle. The stability of a standard difference scheme based on monotone approximations of the problem on a uniform grid is analyzed, and the behavior of discrete solutions in the presence of perturbations is examined. With an increase in the number of grid nodes, this scheme does not converge -uniformly in the maximum norm, but only conditional convergence takes place. When the solution of the difference scheme converges, which occurs if N ₁ ^-1 N ₂ ^-1 ? ε, where N ₁ and N ₂ are the numbers of grid intervals in x and y, respectively, the scheme is not -uniformly well-conditioned or ε-uniformly stable to data perturbations in the grid problem and to computer perturbations. For the standard difference scheme in the presence of data perturbations in the grid problem and/or computer perturbations, conditions imposed on the “parameters” of the difference scheme and of the computer (namely, on ε, N ₁,N ₂, admissible data perturbations in the grid problem, and admissible computer perturbations) are obtained that ensure the convergence of the perturbed solutions as N ₁,N ₂ → ∞, ε ∈ (0,1]. The difference schemes constructed in the presence of the indicated perturbations that converges as N ₁,N ₂ → ∞ for fixed ε, ε ∈ (0,1, is called a computer difference scheme. Schemes converging ε-uniformly and conditionally converging computer schemes are referred to as reliable schemes. Conditions on the data perturbations in the standard difference scheme and on computer perturbations are also obtained under which the convergence rate of the solution to the computer difference scheme has the same order as the solution of the standard difference scheme in the absence of perturbations. Due to this property of its solutions, the computer difference scheme can be effectively used in practical computations.     

A method for finding coefficients of a quasilinear hyperbolic equation

The inverse problem of finding the coefficients q(s) and p(s) in the equation u _tt = a ² u _xx + q(u)u _t ? p(u)u _x is investigated. As overdetermination required in the inverse setting, two additional conditions are set: a boundary condition and a condition with a fixed value of the timelike variable. An iteration method for solving the inverse problem is proposed based on an equivalent system of integral equations of the second kind. A uniqueness theorem and an existence theorem in a small domain are proved for the inverse problem to substantiate the convergence of the algorithm.