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.     

The numerical analysis of two linearized difference schemes for the Benjamin–Bona–Mahony–Burgers equation

In the article, two linearized finite difference schemes are proposed and analyzed for the Benjamin–Bona–Mahony–Burgers (BBMB) equation. For the construction of the two-level scheme, the nonlinear term is linearized via averaging k and k + 1 floor, we prove unique solvability and convergence of numerical solutions in detail with the convergence order O(τ² + h²) . For the three-level linearized scheme, the extrapolation technique is utilized to linearize the nonlinear term based on ψ function. We obtain the conservation, boundedness, unique solvability and convergence of numerical solutions with the convergence order O(τ² + h²) at length. Furthermore, extending our work to the BBMB equation with the nonlinear source term is considered and a Newton linearized method is inserted to deal with it. The applicability and accuracy of both schemes are demonstrated by numerical experiments.     

Smooth Solution of the Compressible Navier–Stokes Equations in an Unbounded Domain with Inflow Boundary Condition

The barotropic compressible Navier–Stokes equations in an unbounded domain are studied. We prove the unique existence of the solution (u, p) of the system (1.1) in the Sobolev spaceH^{k + 3} × H^{k + 2}provided that the derivatives of the data of the problem are sufficiently small, wherek ≥ 0 is any integer. The proof follows from an analysis of the linearized problem, the solvability of the continuity equation, and the Schauder fixed point theory. Similar smoothness results are obtained for a linearized form of (1.1).     

The stability and convergence of two linearized finite difference schemes for the nonlinear epitaxial growth model

The numerical simulation of the dynamics of the molecular beam epitaxy (MBE) growth is considered in this article. The governing equation is a nonlinear evolutionary equation that is of linear fourth order derivative term and nonlinear second order derivative term in space. The main purpose of this work is to construct and analyze two linearized finite difference schemes for solving the MBE model. The linearized backward Euler difference scheme and the linearized Crank‐Nicolson difference scheme are derived. The unique solvability, unconditional stability and convergence are proved. The linearized Euler scheme is convergent with the convergence order of O(τ + h²) and linearized Crank‐Nicolson scheme is convergent with the convergence order of O(τ² + h²) in discrete L²‐norm, respectively. Numerical stability with respect to the initial conditions is also obtained for both schemes. Numerical experiments are carried out to demonstrate the theoretical analysis. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Solvability of Newton equations in smoothing-type algorithms for the SOCCP

In this paper, we first investigate the invertibility of a class of matrices. Based on the obtained results, we then discuss the solvability of Newton equations appearing in the smoothing-type algorithm for solving the second-order cone complementarity problem (SOCCP). A condition ensuring the solvability of such a system of Newton equations is given. In addition, our results also show that the assumption that the Jacobian matrix of the function involved in the SOCCP is a P₀-matrix is not enough for ensuring the solvability of such a system of Newton equations, which is different from the one of smoothing-type algorithms for solving many traditional optimization problems in ℜⁿ.     

Solvability in weighted Hölder spaces for a problem governing the evolution of two compressible fluids

Local (in time) unique solvability of a problem on the motion of two compressible fluids, one of which has finite volume, is obtained in Hölder spaces of functions with a power-like decay at infinity. After passage to Lagrangian coordinates, we arrive at a nonlinear initial boundary value problem with a given closed interface between the liquids. We establish an existence theorem for this problem on the basis of the solvability of a linearized problem by means of the fixed-point theorem. To obtain estimates and to prove the solvability for the linearized problem, we use the Schauder method and an explicit solution of a model linear problem with a plane interface between the liquids. The results are obtained under some restrictions on the fluid density and viscosities, which mean that the fluids are not much different from each other. Bibliography: 8 titles.To Olga Aleksandrovna Ladyzhenskaya on the occasion of her jubilee__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 295, 2003, pp. 57–89.     

On the global existence and time decay estimates in critical spaces for the Navier–Stokes–Poisson system

We are concerned with the study of the Cauchy problem for the Navier–Stokes–Poisson system in the critical regularity framework. In the case of a repulsive potential, we first establish the unique global solvability in any dimension for small perturbations of a linearly stable constant state. Next, under a suitable additional condition involving only the low frequencies of the data and in the L²‐critical framework (for simplicity), we exhibit optimal decay estimates for the constructed global solutions, which are similar to those of the barotropic compressible Navier–Stokes system. Our results rely on new a priori estimates for the linearized Navier–Stokes–Poisson system about a stable constant equilibrium, and on a refined time‐weighted energy functional.     

Degenerate Linear Evolution Equations with the Riemann–Liouville Fractional Derivative

We study the unique solvability of the Cauchy and Schowalter–Sidorov type problems in a Banach space for an evolution equation with a degenerate operator at the fractional derivative under the assumption that the operator acting on the unknown function in the equation is p-bounded with respect to the operator at the fractional derivative. The conditions are found ensuring existence of a unique solution representable by means of the Mittag-Leffler type functions. Some abstract results are illustrated by an example of a finite-dimensional degenerate system of equations of a fractional order and employed in the study of unique solvability of an initial-boundary value problem for the linearized Scott-Blair system of dynamics of a medium.     

An unconditionally stable linearized difference scheme for the fractional Ginzburg-Landau equation

In this paper, we propose a linearized implicit finite difference scheme for solving the fractional Ginzburg-Landau equation. The scheme, which involves three time levels, is unconditionally stable and second-order accurate in both time and space variables. Moreover, the unique solvability, the unconditional stability, and the convergence of the method in the \(L^{\infty }\)-norm are proved by the energy method and mathematical induction. Compared with the implicit midpoint difference scheme (Wang and Huang J. Comput. Phys. 312, 31–49, 2016), current linearized method generally reduces the computational cost. Finally, numerical results are presented to confirm the theoretical results.     

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.     

A second‐order linearized difference scheme for a strongly coupled reaction‐diffusion system

This article deals with the numerical solution to some models described by the system of strongly coupled reaction–diffusion equations with the Neumann boundary value conditions. A linearized three‐level scheme is derived by the method of reduction of order. The uniquely solvability and second‐order convergence in L₂‐norm are proved by the energy method. A numerical example is presented to demonstrate the accuracy and efficiency of the proposed method. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

On mixed boundary value problems for pseudoparabolic systems

Solvability analysis of mixed boundary value problems for pseudoparabolic systems in a special scale of weighted Sobolev spaces is presented. The class under consideration contains the linearized Navier-Stokes system. It is proved that, choosing the power weight, one can diminish the number of solvability conditions and in some cases obtain unconditional solvability of the boundary value problems.     

A finite difference scheme for solving the undamped Timoshenko beam equations with both ends free

In this paper, a boundary feedback system of a class of non-uniform undamped Timoshenko beam with both ends free is considered. A linearized three-level difference scheme for the Timoshenko beam equations is derived by the method of reduction of order on uniform meshes. The unique solvability, unconditional stability and convergence of the difference scheme are proved by the discrete energy method. The convergence order in maximum norm is of order two in both space and time. The validity of this theoretical analysis is verified experimentally.     

Best mean-quasiconformal mappings

We look for best mean-quasiconformal mappings as extremals of the functional equal to the integral of the square of the functional of the conformality distortion multiplied by a special weight. The mapping inverse to an extremal is an extremal of the same functional. We obtain necessary and sufficient conditions for the Petrovskii ellipticity of the system of Euler equations for an extremal. We prove the local unique solvability of boundary values problems for this system in the 2-dimensional case. In the general case we prove the unique solvability of boundary value problems for the system linearized at the identity mapping.     

Error estimates for MAC-like approximations to the linear Navier-Stokes equations

Summary In this paper a priori error estimates are derived for the discretization error which results when the linear Navier-Stokes equations are solved by a method which closely resembles the MAC-method of Harlow and Welch. General boundary conditions are permitted and the estimates are in terms of the discreteL ² norm. A solvability result is given which also applies to a generalization of the method to the nonlinear case. This generalization is used in the last section to produce a numerical solution to the problem of flow around an obstacle.This work supported in part by Westinghouse Nuclear Energy Systems. Research Report #76-13     

Solvability of the problem of evolution of an isolated volume of viscous,incompressible capillary fluid

The initial boundary-value problem for the Navier-Stokes equation describing the flow of a viscous, incompressible capillary fluid bounded only by a free surface is considered. At the initial time the region occupied by the fluid and the velocity field of the fluid are given. A theorem is formulated regarding the unique solvability of the problem for a finite time interval, and a model linearized problem in a half space is obtained.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Sleklova AN SSSR, Vol. 140, pp. 179–186, 1984.     

A two-grid finite element method for the Allen-Cahn equation with the logarithmic potential

In this paper, we present a two-grid finite element method for the Allen-Cahn equation with the logarithmic potential. This method consists of two steps. In the first step, based on a fully implicit finite element method, the Allen-Cahn equation is solved on a coarse grid with mesh size H. In the second step, a linearized system whose nonlinear term is replaced by the value of the first step is solved on a fine grid with mesh size h. We give the energy stabilities of the traditional finite element method and the two-grid finite element method. The optimal convergence order of the two-grid finite element method in H¹ norm is achieved when the mesh sizes satisfy h = O(H²). Numerical examples are given to demonstrate the validity of the proposed scheme. The results show that the two-grid method can save the CPU time while keeping the same convergence rate.     

Numerical solution of matrix equations of the form X + AX T B = C

A review of numerical methods for solving matrix equations of the form X + AX ^T B = C is given. The methods under consideration were implemented in the Matlab environment. The performances of these methods are compared, including the case where the conditions for unique solvability are “almost” violated.     

Homogenization of the linearized system of elasticity in anisotropic heterogeneous thin cylinders

This paper is the second of the two announced in our Note (Sili A, [16]). It generalizes to the linearized system of elasticity the results of our previous work (Sili, 2000 [15]) on the heat equation. We study the asymptotic analysis, as ? tends to zero, of the solution u^? of the linearized system of elasticity posed on a composite elastic cylindrical domain Ω^? with radius ? and height L. The heterogeneities of the material are assumed to be periodic with a period ? along the cylinder axis and with a period ?² along the sections of the cylinder. It is shown that the limit problem is a system in which appear two entities: the first one (u, v, w) corresponds to the reduction of dimension 3d–1d while the second one (û, v?, ?) takes into account the homogenization process. Moreover, a corrector result is given. Copyright © 2002 John Wiley & Sons, Ltd.     

Two-level linearized and local uncoupled difference schemes for the two-component evolutionary Korteweg-de Vries system

The two-level linearized and local uncoupled spatial second order and compact difference schemes are derived for the two-component evolutionary system of nonhomogeneous Korteweg-de Vries equations. It is shown by the mathematical induction that these two schemes are uniquely solvable and convergent in a discrete L_∞ norm with the convergence order of O(τ² + h²) and O(τ² + h⁴), respectively, where τ and h are the step sizes in time and space. Three numerical examples are given to confirm the theoretical results.