Improved Computer Scheme for a Singularly Perturbed Parabolic Convection–Diffusion Equation

Abstract

For a singularly perturbed parabolic convection–diffusion equation with a perturbation parameter $\varepsilon$, $\varepsilon \in (0,1]$, multiplying the highest-order derivative in the equation, we construct an improved computer difference scheme (with approximation of the first-order spatial derivative in the convective term by the central difference operator) on uniform meshes and study the behavior of discrete solutions in the presence of perturbations in the problem data. When solving such a problem numerically, errors in the grid solution depend on the parameter $\varepsilon$, on the parameters of the difference scheme, and also on the value of perturbations introduced in the process of computations (computer perturbations). For small values of the parameter $\varepsilon$, such errors, in general, significantly exceed the solution itself. For the computer perturbations, the conditions imposed on these admissible perturbations are obtained, under which accuracy of the computer solution in order is the same as for the solution of the unperturbed improved difference scheme, namely, $\mathcal {O}(\varepsilon ^{-2}\,N^{-2}+\,N^{-1}_0)$. As a result, we have been constructed the improved computer difference scheme suitable for practical use.

Appendix

Here we give some a priori estimates for solutions and derivatives in initial–boundary value problem (2), (1) that were used to justify convergence of the scheme under construction. These estimates are derived in a similar way as it was done, for example, in [1]; their complete derivation can be found, for example, in [9].

The solution of problem (2), (1) is represented as a sum of functions:

$$\begin{aligned} u(x,t)= U(x,t)+V(x,t), \ (x,t) \in \overline{G}, \end{aligned}$$

(23)

where U(x, t) and V(x, t) are the regular and singular components of the solution.

When the data of this problem are sufficiently smooth and the compatibility conditions at the corner points are satisfied, for the solution of the problem and its component from (23), the following estimates are valid (see, for example, [1]):

$$\begin{aligned} \Big |\frac{\partial ^{k+k_0}}{\partial x^{k}\partial t^{k_0}}\, u(x,t)\Big |\le M \,\varepsilon ^{-k}, \ (x,t) \in \overline{G}, \ k+2k_0 \le 4; \end{aligned}$$

(24)

$$\begin{aligned}&\Big |\frac{\partial ^{k+k_0}}{\partial x^{k}\partial t^{k_0}}\, U(x,t)\Big |\le M, \\[1ex]&\Big |\frac{\partial ^{k+k_0}}{\partial x^{k}\partial t^{k_0}}\, V(x,t)\Big |\le M\,\varepsilon ^{-k}\,\exp ^{-m{\varepsilon ^{-1}\,x}}, \; (x,t) \in \overline{G}, \ k+2\,k_0 \le 4, \nonumber \end{aligned}$$

(25)

where $m \le \min _{\overline{G}} (b(x,t)/c(x,t))$.