Iterative solution of a singular convection-diffusion perturbation problem

An iterative domain decomposition method is developed to solve a singular perturbation problem. The problem consists of a convection-diffusion equation with a discontinuous (piecewise-constant) diffusion coefficient, and the problem domain is decomposed into two subdomains, on each of which the coefficient is constant. After showing that the boundary value problem is well posed, we indicate a specific numerical implementation of the iterative technique that combines the finite element method on one subdomain with the method of matched asymptotic expansions on the other subdomain. This procedure extends work by Carlenzoli and Quarteroni, which was originally intended for a boundary layer problem with an outer region and an inner region. Our extension carries over to a problem where the domain consists of the outer and inner boundary layer regions plus a region in which the diffusion coefficient is constant and significant in magnitude. An unexpected benefit of our new implementation is its efficiency, which is due to the fact that at each iteration the problem needs to be solved explicitly only on one subdomain. It is only when the final approximation on the entire domain is desired that the matched asymptotic expansions approximation need be computed on the second subdomain. Two-dimensional convergence results and numerical results illustrating the method for a two-dimensional test problem are given.Received: February 12, 2004     

Asymptotic representation of a solution to a singular perturbation linear time-optimal problem

A time-optimal control problem is considered for a linear system with fast and slow variables and smooth geometric constraints on the control. An asymptotic expansion of the optimal time up to the second order of smallness is constructed and validated.     

Multigrid convergence for a singular perturbation problem

So far there has been no analysis of multigrid methods applied to singularly perturbed Dirichlet boundary-value problems. Only for periodic boundary conditions does the Fourier transformation (mode analysis) apply, and it is not obvious that the convergence results carry over to the Dirichlet case, since the eigenfunctions are quite different in the two cases. In this paper we prove a close relationship between multigrid convergence for the easily analysable case of periodic conditions and the convergence for the Dirichlet case.     

On a singular perturbation problem in magnetohydrodynamics

Zusammenfassung Die Strömung einer zähen, inkompressiblen und elektrisch leitenden Flüssigkeit über einen rotationssymmetrischen Körper wird studiert mit Hilfe einer singulären Methode der Störungsrechnung. Eine asymptotische, im ganzen Strömungsfeld gültige Lösung wird gegeben für grosse Hartmann-ZahlenM.Die Resultate ergeben folgendes Strömungsbild: Zwei Totwasser-Bereiche von der LängeO (M) und der BreiteO (1) werden vor und nach dem Körper geformt. Sie sind begrenzt durch eine zylindrische Schubschicht, die vom grössten Durchmesser des Körpers aus parabolisch stromaufwärts und stromabwärts anwächst. In einer Entfernung der GrössenordnungO (M) geht diese Schubschicht in eine Wirbelstrasse über, die sich parabolisch ins Unendliche erstreckt. Die Einzelheiten des Strömungsbildes werden analytisch aufgezeigt. Die Wirbelstrasse wird mit derjenigen der klassischen Navier-Stokes-Theorie verglichen.     

Iterative methods for the solution of a singular control formulation of a GMWB pricing problem

Discretized singular control problems in finance result in highly nonlinear algebraic equations which must be solved at each timestep. We consider a singular stochastic control problem arising in pricing a guaranteed minimum withdrawal benefit (GMWB), where the underlying asset is assumed to follow a jump diffusion process. We use a scaled direct control formulation of the singular control problem and examine the conditions required to ensure that a fast fixed point policy iteration scheme converges. Our methods take advantage of the special structure of the GMWB problem in order to obtain a rapidly convergent iteration. The direct control method has a scaling parameter which must be set by the user. We give estimates for bounds on the scaling parameter so that convergence can be expected in the presence of round-off error. Example computations verify that these estimates are of the correct order. Finally, we compare the scaled direct control formulation to a formulation based on a block version of the penalty method (Huang and Forsyth in IMA J Numer Anal 32:320?C351, 2012). We show that the scaled direct control method has some advantages over the penalty method.     

A note on a parameterized singular perturbation problem

We consider a uniform finite difference method on Shishkin mesh for a quasilinear first-order singularly perturbed boundary value problem (BVP) depending on a parameter. We prove that the method is first-order convergent except for a logarithmic factor, in the discrete maximum norm, independently of the perturbation parameter. Numerical experiments support these theoretical results.     

On a singular perturbation problem involving a ``circular-well' potential

We study the asymptotic behavior, as a small parameter goes to 0, of the minimizers for a variational problem which involves a ``circular-well' potential, i.e., a potential vanishing on a closed smooth curve in . We thus generalize previous results obtained for the special case of the Ginzburg-Landau potential.

     

Resonance and asymptotic solutions of a singular perturbation problem

A singular perturbation problem for a second-order ODE witha pair of singular boundary points and a pair of interior second-orderturning points is studied. Four leading-order uniform approximationsare formally constructed, each is restricted to a region includingone critical point. The neighbouring approximations are formallymatched independently on an overlap domain, yielding an asymptoticapproximation to leading order of the general solution. Twogeneralized formulae for the singular and turning point eigenvaluesthat exhibit the resonance conditions are derived. The resonancecriteria due to the influence of every possible combinationof the critical points are investigated.     

Stability of a finite-difference discretization of a singular perturbation problem

A new higher-order finite-difference scheme is proposed for a linear singularly perturbed convection–diffusion problem in one dimension. It is shown how the theory of inverse-monotone matrices, the Lorenz decomposition in particular, can be applied to the stability analysis of the resulting linear system.     

Fourth order algorithms for a semilinear singular perturbation problem

Fourth order finite-difference algorithms for a semilinear singularly perturbed reaction–diffusion problem are discussed and compared both theoretically and numerically. One of them is the method of Sun and Stynes (1995) which uses a piecewise equidistant discretization mesh of Shishkin type. Another one is a simplified version of Vulanović's method (1993), based on a discretization mesh of Bakhvalov type. It is shown that the Bakhvalov mesh produces much better numerical results. This revised version was published online in June 2006 with corrections to the Cover Date.     

Uniform convergence of discretization error for a singular perturbation problem

Cell-centered discretization of the convection-diffusion equation with large Péclet number Pe is analyzed, in the presence of a parabolic boundary layer. It is shown theoretically how, by suitable mesh refinement in the boundary layer, the accuracy can be made to be uniform in Pe, at the cost of a IogPe increase of the number of grid cells, in the case of upwind discretization. Numerical experiments are presented indicating that this can in practice also be achieved with a Pe-independent number of grid cells, both with upwind and central discretization, and with vertex-centered discretization. © 1996 John Wiley & Sons, Inc.     

Asymptotic expansion of a solution to a singular perturbation optimal control problem on an interval with integral constraint

A control problem for solutions of a boundary value problem for a second-order ordinary differential equation with a small parameter at the second derivative is considered on a closed interval. The control is scalar and subject to integral constraints. We construct a complete asymptotic expansion in powers of the small parameter in the Erdélyi sense.     

Matching and multiscale expansions for a model singular perturbation problem

We consider the Laplace–Dirichlet equation in a polygonal domain which is perturbed at the scale ε near one of its vertices. We assume that this perturbation is self-similar, that is, derives from the same pattern for all values of ε. On the base of this model problem, we compare two different approaches: the method of matched asymptotic expansions and the method of multiscale expansion. We enlighten the specificities of both techniques, and show how to switch from one expansion to the other. To cite this article: S. Tordeux et al., C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Uniform fourth order difference scheme for a singular perturbation problem

Summary The numerical solution of a nonlinear singularly perturbed two-point boundary value problem is studied. The developed method is based on Hermitian approximation of the second derivative on special discretization mesh. Numerical examples which demonstrate the effectiveness of the method are presented.This research was partly supported by NSF and SIZ for Science of SAP Vojvodina through funds made available to the U.S.-Yugoslav Joint Board on Scientific and Technological Cooperation (grants JF 544, JF 799)     

A second-order difference scheme for a parameterized singular perturbation problem

In this paper we consider a singularly perturbed quasilinear boundary value problem depending on a parameter. The problem is discretized using a hybrid difference scheme on Shishkin-type meshes. We show that the scheme is second-order convergent, in the discrete maximum norm, independent of singular perturbation parameter. Numerical experiments support these theoretical results.     

On a singular perturbation problem with two second-order turning points

In this paper, we study the singular perturbation problem

where 0<ε1 is a small positive parameter, p(x) and q(x) are sufficiently smooth and strictly positive functions. The main feature of this equation is that there are two second-order turning points in the interval (0,1). Based on the rigorous results on singular perturbation problems with one second-order turning point in our previous work, we obtain a uniform asymptotic approximation for the general solution of the above equation by means of a matching technique.     

Asymptotic estimates for a solution of a singular perturbation optimal control problem on a closed interval under geometric constraints

An optimal control problem is considered for solutions of a boundary value problem for a second-order ordinary differential equation on a closed interval with a small parameter at the second derivative. The control is scalar and satisfies geometric constraints. General theorems on approximation are obtained. Two leading terms of an asymptotic expansion of the solution are constructed and an error estimate is obtained for these approximations.