Numerical approximations to a singularly perturbed convection-diffusion problem with a discontinuous initial condition

Abstract

A singularly perturbed parabolic problem of convection-diffusion type with a discontinuous initial condition is examined. An analytic function is identified which matches the discontinuity in the initial condition and also satisfies the homogenous parabolic differential equation associated with the problem. The difference between this analytical function and the solution of the parabolic problem is approximated numerically, using an upwind finite difference operator combined with an appropriate layer-adapted mesh. The numerical method is shown to be parameter-uniform. Numerical results are presented to illustrate the theoretical error bounds established in the paper.

Appendix: : Set of singular functions

Below, we construct a set of functions $\{ \psi _{i} \}_{i=0}^{4}$ such that Lψ_i = 0; $\psi _{i} \in C^{i-1+\gamma } (\bar { Q}), \ i \geq 1$ and each function ψ_i is smooth within the open region Q ∖Γ^∗.

Define the two singular functions (see [1, (10)])

$$ \psi_{0}(x,t) := \text{erfc} \left (\frac{d(t)-x}{2\sqrt{\varepsilon t}} \right) ,\quad E(x,t) := e^{-\frac{(x-d(t))^{2}}{4\varepsilon t}}. $$

(20)

Then, we explicitly write out the derivatives of these two functions

$$ \begin{array}{@{}rcl@{}} \frac{\partial \psi_{0}}{\partial x} &=& \frac{1}{\sqrt{\varepsilon \pi t}} E , \quad \frac{\partial E}{\partial x} = \frac{d(t)-x}{2\varepsilon t} E,\\ \varepsilon \frac{\partial^{2} \psi_{0}}{\partial x^{2}} &=& \frac{d(t)-x}{2t\sqrt{\varepsilon \pi t}} E, \quad \varepsilon \frac{\partial^{2} E}{\partial x^{2}} = \left( \frac{(d(t)-x)^{2}}{2\varepsilon t} -1 \right)\frac{ E}{2t},\\\\ \frac{\partial \psi_{0}}{\partial t} &=& \frac{1}{\sqrt{\varepsilon \pi t}} \left( \frac{(d(t)-x)-2ta}{2 t} \right) E, \\ \frac{\partial E}{\partial t} &=& \frac{(d(t)-x)}{2\varepsilon t} \left( \frac{(d(t)-x)}{2t} -a(t)\right) E = \frac{\sqrt{ \pi }(d(t)-x)}{2\sqrt{\varepsilon t}}\frac{\partial \psi_{0}}{\partial t}. \end{array} $$

Hence, Lψ₀ = 0. We now construct the singular function ψ₁(x,t). The function $(d(t)-x) \psi _{0} \in C^{\gamma } (\bar { Q})$, but L((d(t) − x)ψ₀)≠ 0. From the expressions above, one can check that

$$ L ((d(t)-x) \psi_{0}) = L\left( 2\frac{\sqrt{\varepsilon t}}{\sqrt{\pi}} E\right). $$

In addition, $\sqrt {t} E\in C^{\gamma } (\bar { Q})$. Then, we define the continuous function

$$ \psi_{1}(x,t) := (d(t)-x) \psi_{0} -2\frac{\sqrt{\varepsilon t}}{\sqrt{\pi}} E. $$

(21)

The remaining functions are similarly constructed. They are given by

$$ \psi_{i} := (d(t)-x) \psi_{i-1}+2\varepsilon t (i-1) \psi_{i-2},\qquad i=2,3,4; $$

(22)

and they satisfy for i = 1, 2, 3, 4

$$ \begin{array}{@{}rcl@{}} \frac{\partial \psi_{i}}{\partial x} = -i \psi_{i-1},\ L\psi_{i} =0 \quad \text{and} \quad \psi_{i} \in C^{i-1+\gamma} (\bar { Q)}. \end{array} $$

Either side of x = d, we have the Taylor expansions for the initial condition

$$ \phi (x) = {\sum}_{i=0}^{4} \phi^{(i)} (d^{*}) \frac{(x-d)^{i}}{i!} + R_{0}(x),\quad d^{*}= \left\{ \begin{array}{ll} d^{-} \quad \text{ if } x < d \\ d^{+} \quad \text{ if } x>d \end{array}\right.; $$

with R₀(x) ∈ C⁴(0, 1). Hence, we present the following expansion³

$$ u (x,t)= 0.5 {\sum}_{i=0}^{4} [\phi^{(i)} ] (d) \frac{(-1)^{i}}{i!} \psi_{i} (x,t) + R (x,t). $$

(23)

Note that for i = 1, 2, 3, 4

$$ \frac{\partial^{i} \psi_{i}}{\partial x^{i}} = (-1)^{i}i! \psi_{0} \quad \text{and} \quad [ \psi_{0} ] (d)=2, $$

which implies that [u⁽ⁱ⁾](d, 0) = [ϕ⁽ⁱ⁾](d).

Define the paramaterized exponential function

$$ E_{\gamma} (x,t) := e^{-\frac{\gamma (x-d(t))^{2}}{4\varepsilon t}},\qquad 0 < \gamma < 1. $$

Using the inequality $\text {erfc}(z) \leq C e^{-z^{2}} \leq C e^{\gamma ^{2}/4}e^{-\gamma z}, \forall z$ it follows that

$$ \begin{array}{@{}rcl@{}} \left \vert \frac{\partial^{j} }{\partial t^{j} } \psi_{0}(x,t)\right \vert, \left \vert \frac{\partial^{j} }{\partial t^{j} } E(x,t)\right \vert & \leq& C \left( \frac{1}{t} +\frac{1}{\sqrt {\varepsilon t}}\right)^{j} E_{\gamma} (x,t);\quad j=1,2; \end{array} $$

(24a)

$$ \begin{array}{@{}rcl@{}} \left \vert \frac{\partial^{i} }{\partial x^{i} } \psi_{0}(x,t)\right \vert, \left \vert \frac{\partial^{i} }{\partial x^{i} } E(x,t)\right \vert & \leq& C\left( \frac{1}{\sqrt {\varepsilon t}}\right)^{i} E_{\gamma} (x,t), \quad 0\leq i \leq 4; \end{array} $$

(24b)

$$ \begin{array}{@{}rcl@{}} \left \vert \frac{\partial }{\partial t } \psi_{1}(x,t)\right \vert &\leq& C \left( 1 +\sqrt {\frac{\varepsilon}{ t}}\right) E (x,t), \end{array} $$

(24c)

$$ \begin{array}{@{}rcl@{}} \left\vert \frac{\partial^{2} }{\partial t^{2} } \psi_{1}(x,t)\right \vert & \leq &C\left (\frac1{t}+\frac{1}{t}\sqrt{\frac{\varepsilon}{t}}+\frac1{\sqrt{\varepsilon t}} \right) E_{\gamma} (x,t), \end{array} $$

(24d)

$$ \begin{array}{@{}rcl@{}} \left \vert \frac{\partial^{i} }{\partial x^{i} } \psi_{1}(x,t) \right \vert &\leq& C\left( \frac{1}{\sqrt {\varepsilon t}}\right)^{i-1} E_{\gamma} (x,t), \ 0 \leq i \leq 4. \end{array} $$

(24e)

For the next terms, we can also establish the bounds

$$ \left \vert \frac{\partial }{\partial t } \psi_{2}(x,t)\right \vert \leq C,\quad \left \vert \frac{\partial^{2} }{\partial x^{2} } \psi_{2}(x,t)\right \vert \leq C; $$

(25a)

on the second time derivatives

$$ \begin{array}{@{}rcl@{}} \left \vert \frac{\partial^{2} }{\partial t^{2} } \psi_{2}(x,t)\right \vert &\leq& C \varepsilon \left( \frac{1}{t} +\frac{1}{\sqrt {\varepsilon t}}+ \frac{1}{\varepsilon} \right) E_{\gamma} (x,t) +C \\ & \leq& C \left (1+ \frac{\varepsilon}{t} \right) E_{\gamma} (x,t) +C, \end{array} $$

(25b)

$$ \begin{array}{@{}rcl@{}} \left \vert \frac{\partial^{2} }{\partial t^{2} } \psi_{3}(x,t)\right \vert &\leq& C \sqrt{\varepsilon t} \left( 1+ \frac{\varepsilon}{t}\right) E_{\gamma} (x,t) +C\vert d(t) -x \vert , \end{array} $$

(25c)

$$ \begin{array}{@{}rcl@{}} \left \vert \frac{\partial^{2} }{\partial t^{2} } \psi_{4}(x,t)\right \vert &\leq& C \varepsilon E_{\gamma} (x,t)+C(\varepsilon +\vert d(t) -x\vert)^{2}; \end{array} $$

(25d)

on the fourth space derivatives

$$ \left \vert \frac{\partial^{4} }{\partial x^{4} } \psi_{j}(x,t) \right \vert \leq C(\sqrt{\varepsilon t})^{j-4} E_{\gamma} (x,t), \quad j=2,3,4 $$

(25e)

and on the third space derivatives

$$ \left\vert \frac{\partial^{3} }{\partial x^{3} } \psi_{2}(x,t) \right \vert \leq C \frac{1}{\sqrt {\varepsilon t}} E_{\gamma} (x,t), \quad \left \vert \frac{\partial^{3} }{\partial x^{3} } \psi_{3}(x,t)\right \vert , \left \vert \frac{\partial^{3} }{\partial x^{3} } \psi_{4}(x,t) \right\vert \leq C. $$

(25f)