# 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.

## Keywords

Perturbation parameter Singularly perturbed initial-boundary value problem Parabolic convection–diffusion equation Boundary layer Difference scheme on uniform meshes Perturbations in data of the grid problem Computer perturbations Stability of schemes to perturbations Conditioning of schemes Computer difference scheme Maximum norm## Notes

### Acknowledgements

This research was partially supported by the Russian Foundation for Basic Research under grant No. 16-01-00727.

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