Boundary-interior layer interactions in nonlinear singular perturbation theory

Howes, F. A.

doi:10.1090/memo/0203

Cited by 70 publications

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“…Numerical intractability of (1.1). To illustrate the substantial difficulties that the numerical solution of the semilinear problem (1.1) presents when we drop the restrictive assumption b u > 0, we consider an example that is a variant of one appearing in [7].…”

Section: Notation Throughout the Paper C Cmentioning

confidence: 99%

“…The reduced problem b(x, φ) = 0 has four solutions: φ 1 = 0, φ 2 = 1, φ 3 (x) = x + 3/2, and φ 4 (x) = −(x + 3/2). An asymptotic analysis [7,20] shows that if a solution of this problem has an interior layer, then that layer must be centered at a certain point that is O(ε) distant from x = 3/8 and the solution is approximately equal to φ 1 and φ 3 respectively to the left and right of the layer. A standard 3-point difference scheme-see (4.2) below-on an equidistant mesh and on an appropriate Shishkin mesh, each having N intervals, yielded the unstable solutions shown in Figure 2.1.…”

Section: Notation Throughout the Paper C Cmentioning

confidence: 99%

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Stabilised approximation of interior-layer solutions of a singularly perturbed semilinear reaction–diffusion problem

Kopteva

Stynes

2011

Numer. Math.

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Abstract.A semilinear reaction-diffusion two-point boundary value problem, whose secondorder derivative is multiplied by a small positive parameter ε 2 , is considered. It can have multiple solutions. The numerical computation of solutions having interior transition layers is analysed. It is demonstrated that the accurate computation of such solutions is exceptionally difficult.To address this difficulty, we propose an artificial-diffusion stabilization. For both standard and stabilised finite difference methods on suitable Shishkin meshes, we prove existence and investigate the accuracy of computed solutions by constructing discrete sub-and super-solutions. Convergence results are deduced that depend on the relative sizes of ε and N , where N is the number of mesh intervals. Numerical experiments are given in support of these theoretical results. Practical issues in using Newton's method to compute a discrete solution are discussed.

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Section: Notation Throughout the Paper C Cmentioning

confidence: 99%

Section: Notation Throughout the Paper C Cmentioning

confidence: 99%

Stabilised approximation of interior-layer solutions of a singularly perturbed semilinear reaction–diffusion problem

Kopteva

Stynes

2011

Numer. Math.

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“…Significant results were provided by Wasow (1956), O'Malley (1969) and Howes (1978). Wasow (1956) established conditions under which systems of the form…”

Section: Nonlinear Second-order Two-point Boundary-value Problemsmentioning

confidence: 99%

“…O'Malley (1969) considered a systems approach to the problem, and Howes (1978) used the stability of the e = 0 problem to study the existence of boundary-, shock-and corner-layer solutions. We shall again restrict ourselves to a situation where only a single boundary layer exists, so follow Howes (1978, p. 79) and consider his example (E3), …”

Section: Nonlinear Second-order Two-point Boundary-value Problemsmentioning

confidence: 99%

Exponential asymptotics and boundary-value problems: keeping both sides happy at all orders

Howls

2010

Proc. R. Soc. A.

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We introduce templates for exponential-asymptotic expansions that, in contrast to matched asymptotic approaches, enable the simultaneous satisfaction of both boundary values in classes of linear and nonlinear equations that are singularly perturbed with an asymptotic parameter e → 0 + and have a single boundary layer at one end of the interval. For linear equations, the template is a transseries that takes the form of a sliding ladder of exponential scales. For nonlinear equations, the transseries template is a two-dimensional array of exponential scales that tilts and realigns asymptotic balances as the interval is traversed. An exponential-asymptotic approach also reveals how boundary-value problems force the surprising presence of transseries in the linear case and negative powers of e terms in the series beyond all orders in the nonlinear case. We also demonstrate how these transseries can be resummed to generate multiplescales-type approximations that can generate uniformly better approximations to the exact solution out to larger values of the perturbation parameter. Finally, we show for a specific example how a reordering of the terms in the exponential asymptotics can lead to an acceleration of the accuracy of a truncated expansion.

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“…will exhibit an interior layer [5] centered along some curve Γ * := {(q(t), t), t > 0}, which has a hyperbolic tangent profile. In the case of the corresponding Cauchy problem posed on the unbounded domain (x, t) ∈ (−∞, ∞) × (0, ∞) with a smooth initial condition y(x, 0) = g(x), x ∈ (−∞, ∞), there will be an initial phase before the interior layer is fully formed [6].…”

Section: Introductionmentioning

confidence: 99%

A linearised singularly perturbed convection–diffusion problem with an interior layer

O’Riordan

Quinn²

2015

Applied Numerical Mathematics

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A linear time dependent singularly perturbed convection-diffusion problem is examined. The convective coefficient contains an interior layer (with a hyperbolic tangent profile), which in turn induces an interior layer in the solution. A numerical method consisting of a monotone finite difference operator and a piecewise-uniform Shishkin mesh is constructed and analysed. Neglecting logarithmic factors, first order parameter uniform convergence is established.

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Boundary-interior layer interactions in nonlinear singular perturbation theory

Cited by 70 publications

References 0 publications

Stabilised approximation of interior-layer solutions of a singularly perturbed semilinear reaction–diffusion problem

Stabilised approximation of interior-layer solutions of a singularly perturbed semilinear reaction–diffusion problem

Exponential asymptotics and boundary-value problems: keeping both sides happy at all orders

A linearised singularly perturbed convection–diffusion problem with an interior layer

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