Convergence rates toward the travelling waves for a model system of the radiating gas

The present paper is concerned with an asymptotics of a solution to the model system of radiating gas. The previous researches have shown that the solution converges to a travelling wave with a rate (1 + t)^?1/4 as time t tends to infinity provided that an initial data is given by a small perturbation from the travelling wave in the suitable Sobolev space and the perturbation is integrable. In this paper, we make more elaborate analysis under suitable assumptions on initial data in order to obtain shaper convergence rates than previous researches. The first result is that if the initial data decays at the spatial asymptotic point with a certain algebraic rate, then this rate reflects the time asymptotic convergence rate. Precisely, this convergence rate is completely same as the spatial convergence rate of the initial perturbation. The second result is that if the initial data is given by the Riemann data, an admissible weak solution, which has a discontinuity, converges to the travelling wave exponentially fast. Both of two results are proved by obtaining decay estimates in time through energy methods with suitably chosen weight functions. Copyright © 2006 John Wiley & Sons, Ltd.     

The low frequency limit for time-harmonic acoustic waves

The question of convergence of the solution of the exterior Dirichlet boundary value problem of the first order system of linearized acoustics as the frequency ω tends to zero is considered. The particular difficulty of having zero as an eigenvalue is handled by introducing certain scalar characteristics that — if prescribed — imply uniqueness in the limit case ω = 0. Convergence to the static solution in a weighted L₂-space is shown. Eventually the convergence result obtained can be extended to a larger class of boundary value problems of Mathematical Physics due to the structure of the argument.     

Runge-Kutta Methods for the Numerical Solution of Stiff Semilinear Systems

This paper studies the stability and convergence properties of general Runge-Kutta methods when they are applied to stiff semilinear systems y(t) = J(t)y(t) + g(t, y(t)) with the stiffness contained in the variable coefficient linear part.We consider two assumptions on the relative variation of the matrix J(t) and show that for each of them there is a family of implicit Runge-Kutta methods that is suitable for the numerical integration of the corresponding stiff semilinear systems, i.e. the methods of the family are stable, convergent and the stage equations possess a unique solution. The conditions on the coefficients of a method to belong to these families turn out to be essentially weaker than the usual algebraic stability condition which appears in connection with the B-stability and convergence for stiff nonlinear systems. Thus there are important RK methods which are not algebraically stable but, according to our theory, they are suitable for the numerical integration of semilinear problems.This paper also extends previous results of Burrage, Hundsdorfer and Verwer on the optimal convergence of implicit Runge-Kutta methods for stiff semilinear systems with a constant coefficients linear part.     

Numerical Solutions of Stochastic Differential Delay Equations with Jumps

Abstract

In this article, we investigate the strong convergence of the Euler–Maruyama method and stochastic theta method for stochastic differential delay equations with jumps. Under a global Lipschitz condition, we not only prove the strong convergence, but also obtain the rate of convergence. We show strong convergence under a local Lipschitz condition and a linear growth condition. Moreover, it is the first time that we obtain the rate of the strong convergence under a local Lipschitz condition and a linear growth condition, i.e., if the local Lipschitz constants for balls of radius R are supposed to grow not faster than log R.     

Approximation of slow solutions to differential inclusions

To approach a viable solution of a differential inclusion, i.e., staying at any time in a closed convexK, a sufficient condition is given implying the convergence of an approximation sequence defined from the Euler or Runge-Kutta methods applied to a selection process which corresponds to the slowsolution concept. WhenK is smooth, the convergence condition is satisfied. This proves that the method is implementable on a computer for solving, for instance, differentiable equations with a noncontinuous right-hand side. Since the usual best approximation operator is difficult to implement, we introduce a class of quasi-projectors much more suitable for computation.     

Anelastic Approximation as a Singular Limit of the Compressible Navier–Stokes System

Considering compressible Navier–Stokes system in a slab geometry in the regime when both Mach and Froude numbers vanish at the same rate, we study the behavior of corresponding weak solutions, that are known to exist globally-in-time (for large data). We establish their convergence to a solution of the so-called anelastic approximation when the limit flow is stratified, i.e., the limit density depends effectively on the vertical coordinate.     

On the geometric convergence of optimized Schwarz methods with applications to elliptic problems

The Schwarz method can be used for the iterative solution of elliptic boundary value problems on a large domain Ω. One subdivides Ω into smaller, more manageable, subdomains and solves the differential equation in these subdomains using appropriate boundary conditions. Optimized Schwarz Methods use Robin conditions on the artificial interfaces for information exchange at each iteration, and for which one can optimize the Robin parameters. While the convergence theory of classical Schwarz methods (with Dirichlet conditions on the artificial interface) is well understood, the overlapping Optimized Schwarz Methods still lack a complete theory. In this paper, an abstract Hilbert space version of the Optimized Schwarz Method (OSM) is presented, together with an analysis of conditions for its geometric convergence. It is also shown that if the overlap is relatively uniform, these convergence conditions are met for Optimized Schwarz Methods for two-dimensional elliptic problems, for any positive Robin parameter. In the discrete setting, we obtain that the convergence factor ρ(h) varies like a polylogarithm of h. Numerical experiments show that the methods work well and that the convergence factor does not appear to depend on h.     

一类求解抛物型方程侧边值问题的最优滤波方法

该文考虑一类特殊的抛物型方程侧边值问题,即一类含有对流项的非标准逆热传导问题. 给定在x=1处的温度测量值来确定区间(0,1)上的未知解u(x, t). 这是一类不适定问题,即问题的解(如果解存在)不连续依赖于数据.为了求解这一问题, 必须采用某些正则化技巧. 该文给出了一种最优滤波方法, 使得问题的真实解和近似解之间的误差估计达到了Hölder型最优. 同时还证明了问题的解在x=0处的收敛性.     

一类球型区域上变系数反向热传导问题

考虑了一类球型区域上变系数反向热传导问题.这个问题是不适定的,即问题的解（若存在）并不连续依赖于测量数据.构造了投影迭代正则化方法,得到了该反问题的正则近似解,同时给出了在先验和后验参数选取规则下精确解与正则近似解之间的收敛性误差估计.最后,通过数值结果验证了该方法的有效性.     

Convergence of consistent and inconsistent finite difference schemes and an acceleration technique

This paper states and generalizes in part some recent results on finite difference methods for Dirichlet problems in a bounded domain Ω which the author has obtained by himself or with coworkers. After stating a superconvergence property of finite difference solution for the case where the exact solution u belongs to , it is remarked that such a property does not hold in general if . Next, a convergence theorem is given for inconsistent schemes under some assumptions. Furthermore, it is shown that the accuracy of the approximate solution can be improved by a coordinate transformation. Numerical examples are also given.     

Convergence of a large time step generalization of Godunov's method for conservation laws

A natural generalization of Godunov's method for Courant numbers larger than 1 is obtained by handling interactions between neighboring Riemann problems linearly, i.e., by allowing waves to pass through one another with no change in strength or speed. This method is well defined for arbitrarily large Courant numbers and can be written in conservation form. It follows that if a sequence of approximations converges to a limit u(x,t) as the mesh is refined, then u is a weak solution to the system of conservation laws. For scalar problems the method is total variation diminishing and every sequence contains a convergent subsequence. It is conjectured that in fact every sequence converges to the (unique) entropy solution provided the correct entropy solution is used for each Riemann problem. If the true Riemann solutions are replaced by approximate Riemann solutions which are consistent with the conservation law, then the above convergence results for general systems continue to hold.     

A finite algorithm for solving general quadratic problems

Here we propose a global optimization method for general, i.e. indefinite quadratic problems, which consist of maximizing a non-concave quadratic function over a polyhedron inn-dimensional Euclidean space. This algorithm is shown to be finite and exact in non-degenerate situations. The key procedure uses copositivity arguments to ensure escaping from inefficient local solutions. A similar approach is used to generate an improving feasible point, if the starting point is not the global solution, irrespective of whether or not this is a local solution. Also, definiteness properties of the quadratic objective function are irrelevant for this procedure. To increase efficiency of these methods, we employ pseudoconvexity arguments. Pseudoconvexity is related to copositivity in a way which might be helpful to check this property efficiently even beyond the scope of the cases considered here.     

Weak sufficient convergence conditions and applications for newton methods

The famous Newton—Kantorovich hypothesis has been used for a long time as a sufficient condition for the convergence of Newton method to a solution of an equation in connection with the Lipschitz continuity of the Fréchet-derivative of the operator involved. Using Lipschitz and center-Lipschitz conditions we show that the Newton—Kantorovich hypothesis is weakened. The error bounds obtained under our semilocal convergence result are finer and the information on the location of the solution more precise than the corresponding ones given by the dominating Newton— Kantorovich theorem, and under the same hypotheses/computational cost, since the evaluation of the Lipschitz also requires the evaluation of the center-Lipschitz constant. In the case of local convergence we obtain a larger convergence radius than before. This observation is important in computational mathematics and can be used in connection to projection methods and in the construction of optimum mesh independence refinement strategies.     

Convergence of vortex methods for weak solutions to the 2-D euler equations with vortex sheet data

We prove the convergence of vortex blob methods to classical weak solutions for the two-dimensional incompressible Euler equations with initial data satisfying the conditions that the vorticity is a finite Radon measure of distinguished sign and the kinetic energy is locally bounded. This includes the important example of vortex sheets. The result is valid as long as the computational grid size h does not exceed the smoothing blob size ε, i.e., h/ε ≦ C.. ©1995 John Wiley & Sons, Inc.     

The effect of non-optimal bases on the convergence of Krylov subspace methods

Summary There are many examples where non-orthogonality of a basis for Krylov subspace methods arises naturally. These methods usually require less storage or computational effort per iteration than methods using an orthonormal basis (optimal methods), but the convergence may be delayed. Truncated Krylov subspace methods and other examples of non-optimal methods have been shown to converge in many situations, often with small delay, but not in others. We explore the question of what is the effect of having a non-optimal basis. We prove certain identities for the relative residual gap, i.e., the relative difference between the residuals of the optimal and non-optimal methods. These identities and related bounds provide insight into when the delay is small and convergence is achieved. Further understanding is gained by using a general theory of superlinear convergence recently developed. Our analysis confirms the observed fact that in exact arithmetic the orthogonality of the basis is not important, only the need to maintain linear independence is. Numerical examples illustrate our theoretical results.This revised version was published online in June 2005 due to a typesetting mistake in the footnote on page 7.     

Stability of Planar Waves in the Allen–Cahn Equation

We study the asymptotic stability of planar waves for the Allen–Cahn equation on ? ⁿ , where n ≥ 2. Our first result states that planar waves are asymptotically stable under any—possibly large—initial perturbations that decay at space infinity. Our second result states that the planar waves are asymptotically stable under almost periodic perturbations. More precisely, the perturbed solution converges to a planar wave as t → ∞. The convergence is uniform in ? ⁿ . Lastly, the existence of a solution that oscillates permanently between two planar waves is shown, which implies that planar waves are not asymptotically stable under more general perturbations.     

Singularities of solutions to linear,second order analytic elliptic equations in two independent variables I. The completely regular boundary

Two methods are described for the a priori location of singularities of solutions to exterior boundary value problems. One uses an expansion for the solution in a circle centered on a regular exterior point P. A singularity lies on the circle of convergence. The envelope of these circles, generated as P makes a circuit about the closed boundary, circumscribes the singularities. The radius of convergence depends on singularities of the solution u(s) and its normal derivative v(s) on the boundary. The second method employs complex characteristics to relate singularities of the boundary data to real singularities of the solution. Integral equations connecting (y), v(s) and the analytic boundary condition are used to continue the data into the complex s-plane and to locate their singularities. Explicit solution of the integral equations is unnecessary; some nonlinear boundary conditions can be handled.     

Convergence of parallel multistep hybrid methods for singular perturbation problems

Parallel multistep hybrid methods (PHMs) can be implemented in parallel with two processors, accordingly have almost the same computational speed per integration step as BDF methods of the same order with the same stepsize. But PHMs have better stability properties than BDF methods of the same order for stiff differential equations. In the present paper, we give some results on error analysis of A(α)-stable PHMs for the initial value problems of ordinary differential equations in singular perturbation form. Our convergence results are similar to those of linear multistep methods (such as BDF methods), i.e. the convergence orders are equal to their classical convergence orders, and no order reduction occurs. Some numerical examples also confirm our results.     

Steady flows of Jeffrey–Hamel type from the half-plane into an infinite channel. 1. Linearization on an antisymmetric solution

In this paper our objective is to provide physically reasonable solutions for the stationary Navier–Stokes equations in a two-dimensional domain with two outlets to infinity, a semi-strip Π₋ and a half-plane K. The same problem in an aperture domain, i.e. in a domain with two half-plane outlets to infinity, has been studied but only under symmetry restrictions on the data. Here, we assume that the main asymptotic term of the solution takes an antisymmetric form in K and apply the technique of weighted spaces with detached asymptotics, i.e. we use spaces where the functions have prescribed asymptotic forms in the outlets.After first showing that the corresponding Stokes problem admits a unique solution if and only if certain compatibility conditions are satisfied, we write the Navier–Stokes equations as a perturbation of the Stokes problem and the crucial compatibility condition as an algebraic equation by which the flux becomes determined. Assuming that the coefficient of the main (antisymmetric) asymptotic term of the solution in K does not vanish and that the data are sufficiently small, we use a contraction principle to solve the Navier–Stokes system coupled with the algebraic equation.Finally, we discuss the ill-posedness of the Navier–Stokes problem with prescribed flux.     

A Survey of Algebraic Multilevel Iteration (AMLI) Methods

For the solution by preconditioned conjugate gradient methods of symmetric positive definite equations as arising in boundary value problems we consider preconditioning methods of AMLI type. Particular attention is devoted to providing methods of optimal order of computational complexity which in addition promise to be robust, i.e. with a convergence rate which is bounded above independently of size of discretization parameter h, jumps in problem coefficients, and shape of finite elements or, equivalently, anisotropy of problem coefficients. In addition, the computational cost per iteration step must have optimal order.New results on upper bounds of one of the important parameters in the methods, the Cauchy—Bunyakowski—Schwarz constant are given and an algebraic method how to improve its value is presented.This revised version was published online in October 2005 with corrections to the Cover Date.