Multiscale Convection in One Dimensional Singularly Perturbed Convection-Diffusion Problems

Linear singularly perturbed ordinary differential equations of convection diffusion type are considered. The convective coefficient varies in scale across the domain which results in interior layers appearing in areas where the convective coefficient decreases from a scale of order one to the scale of the diffusion coefficient. Appropriate parameter-uniform numerical methods are constructed. Numerical results are given to illustrate the theoretical error bounds established. 1 Continuous problem class In this paper, we examine singularly perturbed ordinary differential equations of the form −εu′′ + aε(x)u + b(x)u = f(x), aε(x) > 0; x ∈ (0, 1), where the magnitude of the convective coefficient aε(x) varies in scale across the domain. Problems of this type may arise when linearizing certain nonlinear singularly perturbed problems or when generating approximations to the solution of a coupled system of singularly perturbed equations. Define the subdomains Ωi, i = 1, 2, 3 of [0, 1] to be Ωi := (di−1, di), 0 = d0 ≤ d1 < d2 ≤ d3 = 1; Ω := ∪i=1Ωi. The points d1, d2 are points where the convective coefficient is of the same order as the diffusion coefficient ε in the following class of singularly perturbed problems: find u ∈ C(Ω) ∩ C(Ω̄) such that Lεu := −εu′′ + aε(x)u + b(x)u = f(x), x ∈ Ω, (1.1a) u(0) = u(1) = 0, (1.1b) b(x) ≥ β > 0, x ∈ Ω, (1.1c) 1 = ‖a‖ ≥ aε(x) > ε, x ∈ Ω2, (1.1d) ε ≥ aε(x) ≥ αε > 0, |aε(x)| ≤ Cε, x ∈ Ω1 ∪Ω3, (1.1e) where 0 < ε ≤ 1 is a singular perturbation parameter and aε, f, b are smooth. The points d1, d2 are assumed to be independent of the singular perturbation parameter ε. For all x ∈ [d1, d2], define the limiting function a0(x) by a0(x) := lim ε→0 aε(x), x 6= d1, d2; a0(d1) := lim x→d+1 a0(x), and a0(d2) := lim x→d2 a0(x). Observe that a0(x) ≡ 0, x ∈ [0, d1)∪ (d2, 1], a0(x) > 0, x ∈ (d1, d2). The different cases of a0(x) being either continuous or discontinuous at the point d1 will be examined. The nature of the growth (decay) of the convective coefficient in the vicinity of the point d1(d2) will be restricted by the following constraints on the problem class: assume also that for a given constant 0 < θ ≤ 0.5 independent of ε, there exists two points γ1, γ2 ∈ Ω2, γ1 < γ2 such that ∫ d2 γ2 aε(t)dt ≥ θ(d2 − d1) > 0; (1.1f) aε(x) ≥ θ > 0, |aε(x)| ≤ C, γ1 ≤ x ≤ γ2; (1.1g) εaε(x) ≤ θ1aε(x), θ1 < 1, x ∈ (d1, γ1); aε(x) ≤ 0, x ∈ (γ2, d2); (1.1h) aε(x)− a0(x) = 3 ∑ i=1 ξi(x; ε), x ∈ Ω2; (1.1i) where for k = 0, 1 and all x ∈ Ω2 |ξ1(x)|k ≤ Cε−k/2e−θ1(x−d1)/ √ , |ξ2(x)|k ≤ Cε−ke−θ(d2−x)/ε, |ξ3|k ≤ Cε. Note that γi, θi, i = 1, 2 are independent of ε. Specific choices for the convective coefficient are taken in the test examples examined in §5. In this paper, our interest is focused on the interior layers appearing in the interior region Ω2. To exclude the appearance of reaction-diffusion type layers in the region Ω1 ∪Ω3, we impose the following restriction on the forcing term f |f(x)| ≤ C1a0(x), x ∈ Ω̄. (1.1j) As f ∈ C(0, 1) and |f(x)| ≤ Ca0(x), then (1.1j) implies that f(x) ≡ 0, x ∈ Ω̄1 ∪ Ω̄3. To exclude any layer emerging in the vicinity of the point d1 and consequently being convected throughout the domain, we impose the additional constraint on the forcing term, that f(x) ≡ 0, x ∈ (d1, 2γ1]. (1.1k) A significant effect of interest in this problem is that the problem is singularly perturbed only in the subdomain Ω2 ⊂ Ω. Within the literature on singularly perturbed convection-diffusion equations, it is normally assumed that a(x) ≥ α > 0 everywhere. In this paper, we examine the effect of 0 < aε(x) ≤ Cε in an O(1)-neighbourhood of the outflow boundary point x = 1.