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｜）.     

Semidiscrete and asymptotic approximations for the nonstationary radiative–conductive heat transfer problem in a periodic system of grey heat shields

We consider semidiscrete and asymptotic approximations to a solution to the nonstationary nonlinear initial-boundary-value problem governing the radiative–conductive heat transfer in a periodic system consisting of n grey parallel plate heat shields of width ε = 1/n, separated by vacuum interlayers. We study properties of special semidiscrete and homogenized problems whose solutions approximate the solution to the problem under consideration. We establish the unique solvability of the problem and deduce a priori estimates for the solutions. We obtain error estimates of order O( ?{e} ) O\left( {\sqrt {\varepsilon } } \right) and O(ε) for semidiscrete approximations and error estimates of order O( ?{e} ) O\left( {\sqrt {\varepsilon } } \right) and O(ε ^3/4) for asymptotic approximations. Bibliography: 9 titles.     

Almost global wellposedness of the 2-D full water wave problem

We consider the problem of global in time existence and uniqueness of solutions of the 2-D infinite depth full water wave equation. It is known that this equation has a solution for a time period [0,T/ε] for initial data of the form ε Ψ, where T depends only on Ψ. In this paper, we show that for such data there exists a unique solution for a time period [0,e ^T/ε ]. This is achieved by better understandings of the nature of the nonlinearity of the full water wave equation. Financial support provided in part by NSF grant DMS-0400643.     

Approximation of singularly perturbed parabolic equations in unbounded domains subject to piecewise smooth boundary conditions in the case of solutions that grow at infinity

The initial-boundary value problem in a domain on a straight line that is unbounded in x is considered for a singularly perturbed reaction-diffusion parabolic equation. The higher order derivative in the equation is multiplied by a parameter ɛ², where ɛ ∈ (0, 1]. The right-hand side of the equation and the initial function grow unboundedly as x → ∞ at a rate of O(x ²). This causes the unbounded growth of the solution at infinity at a rate of O(Ψ(x)), where Ψ(x) = x ² + 1. The initialboundary function is piecewise smooth. When ɛ is small, a boundary and interior layers appear, respectively, in a neighborhood of the lateral part of the boundary and in a neighborhood of the characteristics of the reduced equation passing through the discontinuity points of the initial function. In the problem under examination, the error of the grid solution grows unboundedly in the maximum norm as x → ∞ even for smooth solutions when ɛ is fixed. In this paper, the proximity of solutions of the initial-boundary value problem and its grid approximations is considered in the weighted maximum norm ∥·∥ ^w with the weighting function Ψ⁻¹(x); in this norm, the solution of the initial-boundary value problem is ɛ-uniformly bounded. Using the method of special grids that condense in a neighborhood of the boundary layer or in neighborhoods of the boundary and interior layers, special finite difference schemes are constructed and studied that converge ɛ-uniformly in the weighted norm. It is shown that the convergence rate considerably depends on the type of nonsmoothness in the initial-boundary conditions. Grid approximations of the Cauchy problem with the right-hand side and the initial function growing as O(Ψ(x)) that converge ɛ-uniformly in the weighted norm are also considered.     

Rank-two update algorithms for the minimum volume enclosing ellipsoid problem

We consider the problem of computing a (1+ε)-approximation to the minimum volume enclosing ellipsoid (MVEE) of a given set of m points in R ⁿ . Based on the idea of sequential minimal optimization (SMO) method, we develop a rank-two update algorithm. This algorithm computes an approximate solution to the dual optimization formulation of the MVEE problem, which updates only two weights of the dual variable at each iteration. We establish that this algorithm computes a (1+ε)-approximation to MVEE in O(mn ³/ε) operations and returns a core set of size O(n ²/ε) for ε∈(0,1). In addition, we give an extension of this rank-two update algorithm. Computational experiments show the proposed algorithms are very efficient for solving large-scale problem with a high accuracy.     

Algorithms of Intrinsic Complexity for Point Searching in Compact Real Singular Hypersurfaces

For a real square-free multivariate polynomial F, we treat the general problem of finding real solutions of the equation F=0, provided that the real solution set {F=0}_ℝ is compact. We allow that the equation F=0 may have singular real solutions. We are going to decide whether this equation has a non-singular real solution and, if this is the case, we exhibit one for each generically smooth connected component of {F=0}_ℝ. We design a family of elimination algorithms of intrinsic complexity which solves this problem. In the worst case, the complexity of our algorithms does not exceed the already known extrinsic complexity bound of (nd) ^O(n) for the elimination problem under consideration, where n is the number of indeterminates of F and d its (positive) degree. In the case that the real variety defined by F is smooth, there already exist algorithms of intrinsic complexity that solve our problem. However, these algorithms cannot be used in case when F=0 admits F-singular real solutions.     

A theorem of the Phragmén-Lindelöf type for solutions of an evolution equation of the second order with respect to time variable

We consider a solution u(x, t) of the general linear evolution equation of the second order with respect to time variable given on the ball Π(T) = {(x,t): xε R ⁿ, t ε [0, T]} and study the dependence of the behavior of this solution on the behavior of the functions at infinity. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 5, pp. 724–731, May, 1998.     

Homogenization of the diffusion equation with nonlinear flux condition on the interior boundary of a perforated domain — the influence of the scaling on the nonlinearity in the effective sink-source term

In this paper, we study the asymptotic behavior of solutions u _ε of the initial boundary value problem for parabolic equations in domains W_e ì \mathbbRⁿ {\Omega_\varepsilon } \subset {\mathbb{R}^n} , n ≥ 3, perforated periodically by balls with radius of critical size ε ^α , α = n/(n − 2), and distributed with period ε. On the boundary of the balls a nonlinear third boundary condition is imposed. The weak convergence of the solutions u _ε to the solution of an effective equation is given. Furthermore, an improved approximation for the gradient of the microscopic solutions is constructed, and a corrector result with respect to the energy norm is proved.     

Complex ordering and stochastic oscillations in a class of reaction-diffusion systems with small diffusion

Summary We consider four models of partial differential equations obtained by applying a generalization of the method of normal forms to two-component reaction-diffusion systems with small diffusionu _t=εDu _xx+(A+εA ₁)u+F(u),u ∈ ℝ². These equations (quasinormal forms) describe the behaviour of solutions of the original equation forε → 0. One of the quasinormal forms is the well-known complex Ginzburg-Landau equation. The properties of attractors of the other three equations are considered. Two of these equations have an interesting feature that may be called asensitivity to small parameters: they contain a new parameterϑ(ε)=−(aε ^−1/2 mod 1) that influences the behaviour of solutions, but changes infinitely many times whenε → 0. This does not create problems in numerical analysis of quasinormal forms, but makes numerical study of the original problem involvingε almost impossible.     

On a Global Complexity Bound of the Levenberg-Marquardt Method

In this paper, we investigate a global complexity bound of the Levenberg-Marquardt method (LMM) for the nonlinear least squares problem. The global complexity bound for an iterative method solving unconstrained minimization of φ is an upper bound to the number of iterations required to get an approximate solution, such that ‖∇φ(x)‖≤ε. We show that the global complexity bound of the LMM is O(ε ⁻²).