Convergence of Vortex Methods for Initial Boundary Value Problems

In this paper we study the vortex methods for the Euler equation of incompressible flow with initial and boundary conditions. Second order extrapolation scheme is applied to calculate the motion of vortex blobs near boundary. Under some corresponding conventional hypotheses second order convergence result is proved, which achieves the same precision as that of the pure initial value problems and can be generalized to any higher order without difficulty. The convergence of the vortex-in-cell method where the stream functions are approximated by the finite element method is also proved.     

A class of nonlinear singularly perturbed initial boundary value problems for reaction diffusion equations with boundary perturbation

A class of nonlinear initial boundary value problems for reaction diffusion equations with boundary perturbation is considered. Under suitable conditions and using the theory of differential inequalities the asymptotic solution of the initial boundary value problems is studied.     

Construction of Solutions and L^1-error Estimates of Viscous Methods for Scalar Conservation Laws with Boundary

This paper is concerned with an initial boundary value problem for strictly convex conservation laws whose weak entropy solution is in the piecewise smooth solution class consisting of finitely many discontinuities. By the structure of the weak entropy solution of the corresponding initial value problem and the boundary entropy condition developed by Bardos-Leroux Nedelec, we give a construction method to the weak entropy solution of the initial boundary value problem. Compared with the initial value problem, the weak entropy solution of the initial boundary value problem includes the following new interaction type： an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary. According to the structure and some global estimates of the weak entropy solution, we derive the global L^1-error estimate for viscous methods to this initial boundary value problem by using the matching travelling wave solutions method. If the inviscid solution includes the interaction that an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary, or the inviscid solution includes some shock wave which is tangent to the boundary, then the error of the viscosity solution to the inviscid solution is bounded by O（ε^1/2） in L^1-norm; otherwise, as in the initial value problem, the L^1-error bound is O（ε｜ In ε｜）.     

二维涡度方程chebyshev-谱差分方法的误差估计

We construct a Chebyshev spectral-difference scheme for solving two-dimenslonal vorticity equation with semi-homogeneous boundary conditions. We vigorously analyse the errors produced by initial errors and boundary value errors. The energy method is the main method used for errors estimation in this paper.     

The nonlinear nonlocal singularly perturbed problems for reaction diffusion equations with a boundary perturbation

The nonlinear nonlocal singularly perturbed initial boundary value problems for reaction diffusion equations with a boundary perturbation is considered. Under suitable conditions, the outer solution of the original problem is obtained. Using the stretched variable, the composing expansion method and the expanding theory of power series the initial layer is constructed. And then using the theory of differential inequalities the asymptotic behavior of solution for the initial boundary value problems is studied. Finally the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation are discussed.     

具有边界摄动的非线性非局部反应扩散方程的奇摄动问题

A class of nonlinear nonlocal singularly perturbed Robin initial boundary value problems for reaction diffusion equations with boundary perturbation is considered. Under suitable conditions, firstly, the outer solution of the original problem is obtained; secondly, by using the stretched variable, the composing expansion method and the expanding theory of power series, the initial layer is constructed; and finally, by using the theory of differential inequalities the asymptotic behavior of solutions for initial boundary value problems is studied, and including some relational inequalities the existence and uniqueness of solutions for the original problem and the uniformly valid asymptotic estimation are discussed.     

SINGULARLY PERTURBED SOLUTION TO SEMILINEAR HIGHER ORDER REACTION DIFFUSION EQUATIONS WITH TWO PARAMETERS

A boundary value problem for semilinear higher order reaction diffusion equa-tions with two parameters is considered. Under suitable conditions, by the theory of differential inequalities, the existence and asymptotic behavior of solutions to initial boundary value problem are studied.     

THE NAVIER-STOKES EQUATIONS WITH THE KINEMATIC AND VORTICITY BOUNDARY CONDITIONS ON NON-FLAT BOUNDARIES

We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a general domain in R^n with compact and smooth boundary, subject to the kinematic and vorticity boundary conditions on the non-flat boundary. We observe that, under the nonhomogeneous boundary conditions, the pressure p can be still recovered by solving the Neumann problem for the Poisson equation. Then we establish the well-posedness of the unsteady Stokes equations and employ the solution to reduce our initial-boundary value problem into an initial-boundary value problem with absolute boundary conditions. Based on this, we first establish the well-posedness for an appropriate local linearized problem with the absolute boundary conditions and the initial condition （without the incompressibility condition）, which establishes a velocity mapping. Then we develop apriori estimates for the velocity mapping, especially involving the Sobolev norm for the time-derivative of the mapping to deal with the complicated boundary conditions, which leads to the existence of the fixed point of the mapping and the existence of solutions to our initial-boundary value problem. Finally, we establish that, when the viscosity coefficient tends zero, the strong solutions of the initial-boundary value problem in R^n（n ≥ 3） with nonhomogeneous vorticity boundary condition converge in L^2 to the corresponding Euler equations satisfying the kinematic condition.     

MATHEMATICAL PROBLEMS ON THE FLUID-SOLUTE-HEAT FLOW THROUGH POROUS MEDIA Ⅰ. UNSATURATED CASE

A degenerate parabolic system arising from the fluid-solute-heat flow through unsaturatedporous media is considered. The existence of weak solutions to the initial boundary value problemof this system is esteblished by parabolic regularizaton. The regularity is proved as well, i. e. theweak solutions satisfy the equations and initial boundary conditions pointwise in the region wherethe moisture content is positive.     

Global Weak Solutions of Initial Boundary Value Problem for Boltzmann－Poisson System with Absorbing Boundary

This paper deals with the initial boundary value problem for the Boltzmann-Poisson system, which arises in semiconductor physics, with absorbing boundary. The global existence of weak solutions is proved by using the stability of velocity averages and the compactness results on L1-theory under weaker conditons on initial boundary values.     

An Artificial Boundary Condition for the Vortex Movements in Two Dimensions

1 Motion of vortices and cloud in cell method The motion of incompressible inviscid ?ow in two dimensions can be described by the equations ?u ?t (u ?) ρ1 ?P = f (1) ?u = 0 , (2) where u = (u, v), ρ, P , and f = (f1, f2) denote ?uid velocity, densit…     

On Vortex Methods for Initial Boundary Value Problems

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On non‐stationary viscous incompressible flow through a cascade of profiles

The paper deals with theoretical analysis of non‐stationary incompressible flow through a cascade of profiles. The initial‐boundary value problem for the Navier–Stokes system is formulated in a domain representing the exterior to an infinite row of profiles, periodically spaced in one direction. Then the problem is reformulated in a bounded domain of the form of one space period and completed by the Dirichlet boundary condition on the inlet and the profile, a suitable natural boundary condition on the outlet and periodic boundary conditions on artificial cuts. We present a weak formulation and prove the existence of a weak solution. Copyright © 2006 John Wiley & Sons, Ltd.     

Temporal boundary value problems in interfacial fluid dynamics

We study problems in interfacial fluid dynamics which do not have well-posed initial value problems. We prove existence of solutions for these problems by considering instead boundary value problems, where boundary data is specified at two different times. We develop a general framework, for problems on the real line and for problems which are spatially periodic. A variety of boundary conditions are considered, including Dirichlet, Neumann and mixed conditions. The framework is applied to two specific problems from interfacial fluid dynamics: a family of generalizations of the Boussinesq equations developed by Bona, Chen and Saut, and the vortex sheet.     

Convergence of Vortex with Boundary Element Methods

1.IntroductionThepaperwrittenbyChorin[61in1973wasthebasisofthevortxmethods.Hedividednumericalprogramintothreesteps:thefirststepistosolvetheEulerequationwiththevortexmethod,wherethevelocityfliediscomputedfromthevorticityfieldwiththeboundaryelementmethod;thesecondstepistoproducethevorticityontheboundary;thethirdstepistosimulatediffusionwithrandommethod.Itisverydifficultthattobuildthefullymathematictheoreyofvortexmethods.NonecangettheconvergenceofChorin'salgorithmnow.In1978,Chorin,Hughes,McCr…     

Numerical solution to a linearized KdV equation on unbounded domain

Exact absorbing boundary conditions for a linearized KdV equation are derived in this paper. Applying these boundary conditions at artificial boundary points yields an initial‐boundary value problem defined only on a finite interval. A dual‐Petrov‐Galerkin scheme is proposed for numerical approximation. Fast evaluation method is developed to deal with convolutions involved in the exact absorbing boundary conditions. In the end, some numerical tests are presented to demonstrate the effectiveness and efficiency of the proposed method.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

Blowup of solutions of the three-dimensional Rosenau-Burgers equation

We consider the initial boundary value problem for the well-known three-dimensional Rosenau-Burgers equation in the cylinder \((0,L) \otimes \mathbb{S}\) (where \(\mathbb{S} \subset \mathbb{R}^2\)) for some boundary conditions. Using the test-function method, we obtain the result on the blowup of solutions of this initial boundary value problem during a finite time. This is one of the first results in the “blowup” direction for this equation.     

Optimal boundary control by an elastic force at one end and a displacement at the other end for an arbitrary sufficiently large time interval in the string vibration problem

We consider a boundary value problem for the wave equation with given initial conditions and with boundary conditions of the second kind at one end of the string and boundary conditions of the first kind at the other end of the string. We assume the boundary conditions to ensure that the solution of the problem (in the class of generalized functions) satisfying the initial conditions at the initial time t = 0 satisfies given terminal conditions at the terminal time t = T. We clarify the relationship between the functions µ(t) and ν(t) in the boundary conditions and the given functions specifying the initial and terminal states. We obtain closed-form analytic expressions for the functions µ(t) and ν(t) minimizing the boundary energy functional.     

Artificial boundary conditions for Petrovsky systems of second order in exterior domains and in other domains of conical type

The approximation of solutions to boundary value problems on unbounded domains by those on bounded domains is one of the main applications for artificial boundary conditions. Based on asymptotic analysis, here a new method is presented to construct local artificial boundary conditions for a very general class of elliptic problems where the main asymptotic term is not known explicitly. Existence and uniqueness of approximating solutions are proved together with asymptotically precise error estimates. One class of important examples includes boundary value problems for anisotropic elasticity and piezoelectricity. Copyright © 2004 John Wiley & Sons, Ltd.     

Analysis of a Large Eddy Simulation model based on anisotropic filtering

We consider a new Large Eddy Simulation model, derived with the introduction of suitable horizontal (anisotropic) differential filters. One main advantage of this filtering is that, for channel flows, there is no need for artificial boundary conditions. Hence, we can deal with some realistic problems, equipped with Dirichlet boundary conditions, in special bounded domains (at least those bounded only in one direction). Recent numerical results for a similar model, based on a derivation with wave-number asymptotics, are also recalled. After a detailed analysis of the properties of the differential filter, we prove that the resulting initial–boundary value problem is well-posed in suitable anisotropic Sobolev spaces, giving a strong mathematical support to the model we propose. Some remarks on higher-accuracy Approximate Deconvolution Models are also given in the last section.