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

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.

1. As in [3], we define the space \({\mathcal C}^{0+\gamma }(D )\), where D ⊂R² is an open set, as the set of all functions that are Hölder continuous of degree γ ∈ (0,1) with respect to the metric ∥⋅∥, where for all \(\textbf {p}_{i}=(x_{i},t_{i}), \in \mathbf {R}^{2}, i=1,2; \ \Vert \textbf {p}_{1}- \textbf {p}_{2} \Vert ^{2} = (x_{1}-x_{2})^{2} + \vert t_{1} -t_{2} \vert \). For f to be in \({\mathcal C}^{0+\gamma }(D ) \) the following semi-norm needs to be finite

   $$ \lceil f \rceil_{0+\gamma , D} = \sup_{\textbf{p}_{1} \neq \textbf{p}_{2}, \ \textbf{p}_{1}, \textbf{p}_{2}\in D} \frac{\vert f(\textbf{p}_{1}) - f(\textbf{p}_{2}) \vert}{\Vert \textbf{p}_{1}- \textbf{p}_{2} \Vert^{\gamma}} . $$

   The space \({\mathcal C}^{n+ \gamma }(D ) \) is defined by

   $$ {\mathcal C}^{n+\gamma }(D ) = \left \{ z : \frac{\partial^{i+j} z}{ \partial x^{i} \partial t^{j} } \in {\mathcal C}^{0+\gamma }(D), \ 0 \leq i+2j \leq n \right \}, $$

   and ∥⋅∥_n+γ, ⌈⋅⌉_n+γ are the associated norms and semi-norms.

2. We use the following notation for the finite difference approximations of the derivatives:

   $$ \begin{array}{@{}rcl@{}} D^{-}_{t} Y (x_{i},t_{j}) := \frac{Y(x_{i},t_{j})-Y(x_{i},t_{j-1})}{k}, \quad D^{-}_{x} Y(x_{i},t_{j}) :=\frac{Y(x_{i},t_{j})-Y (x_{i-1},t_{j})}{h_{i}}, \\ D^{+}_{x} Y (x_{i},t_{j}) :=\frac{Y(x_{i+1},t_{j})-Y(x_{i},t_{j})}{h_{i+1}}, \ {\delta^{2}_{x}} Y(x_{i},t_{j}) := \frac{2}{h_{i}+h_{i+1}}(D_{x}^{+}Y(x_{i},t_{j})-D^{-}_{x} Y(x_{i},t_{j})). \end{array} $$

3. Bobisud [1] constructs an asymptotic expansion for the solution of (??) of the form \(u=y+R,\ \Vert R \Vert \leq C \sqrt {\varepsilon }\). The construction of y in [1, (10)] identifies the leading term ψ₀(x,t) in (23). However, for the numerical analysis, this basic asymptotic expansion will not suffice as parameter explicit bounds on the partial derivatives of the remainder are required to establish an error bound on any numerical approximations.

Authors and Affiliations

1. Department of Applied Mathematics, University of Zaragoza, Zaragoza, Spain

   J. L. Gracia

2. School of Mathematical Sciences, Dublin City University, Dublin, 9, Ireland

   E. O’Riordan

Corresponding author

Correspondence to J. L. Gracia.

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This research was partially supported by the Institute of Mathematics and Applications (IUMA), the projects PID2019-105979GB-I00 and PGC2018-094341-B-I00 and the Diputación General de Aragón (E24-17R).

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)])

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

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

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

The remaining functions are similarly constructed. They are given by

and they satisfy for i = 1, 2, 3, 4

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

with R₀(x) ∈ C⁴(0, 1). Hence, we present the following expansion^{Footnote 3}

Note that for i = 1, 2, 3, 4

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

Define the paramaterized exponential function

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

For the next terms, we can also establish the bounds

on the second time derivatives

on the fourth space derivatives

and on the third space derivatives

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