# Difference Schemes of High Accuracy Order on Uniform Grids for a Singularly Perturbed Parabolic Reaction-Diffusion Equation

## Abstract

For a singularly perturbed parabolic reaction-diffusion equation with a perturbation parameter \(\varepsilon\) (\(\varepsilon \in (0,1]\)) multiplying the highest-order derivative, we consider a technique to construct \(\varepsilon\)-uniformly convergent in the maximum norm difference schemes of higher accuracy order on uniform grids. In constructing such schemes, we use the solution decomposition method, in which grid approximations of the regular and singular components in the solution are considered on uniform grids. Increasing of the convergence rate of the scheme constructed with improved accuracy of order \(\mathcal{O}\left (N^{-4}\,\ln ^{4}\,N + N_{0}^{-2}\right )\), where *N* and *N*_{0} are the number of nodes in the meshes in *x* and *t*, respectively, is achieved using a Richardson extrapolation technique applied to the regular and singular components. In the proposed Richardson technique, when constructing embedded grids we use most dense grids as main grids. This approach allows us to construct schemes that converge \(\varepsilon\)-uniformly in the maximum norm at the rate \(\mathcal{O}\left (N^{-6}\,\ln ^{6}\,N + N_{0}^{-3}\right )\) and higher.

## Notes

### Acknowledgements

This research was partially supported by the Russian Foundation for Basic Research under grant No.13-01-00618.

## References

- 1.Kalitkin, N.N.: Numerical Methods. Nauka, Moscow (in Russian) (1978)Google Scholar
- 2.Marchuk, G.I.: Methods of Numerical Mathematics, 2nd edn. Springer, New York/Berlin (1982)CrossRefzbMATHGoogle Scholar
- 3.Samarskii, A.A.: The Theory of Difference Schemes. Marcel Dekker, New York (2001)CrossRefzbMATHGoogle Scholar
- 4.Shishkin, G.I.: Discrete Approximations of Singularly Perturbed Elliptic and Parabolic Equations. Russian Academy of Sciences, Ural Section, Ekaterinburg (in Russian) (1992)Google Scholar
- 5.Shishkin, G.I.: Robust novel high-order accurate numerical methods for singularly perturbed convection-diffusion problems. Math. Mod. Anal.
**10**, 393–412 (2005)MathSciNetzbMATHGoogle Scholar - 6.Shishkin, G.I.: Richardson’s method for increasing the accuracy of difference solutions of singularly perturbed elliptic convection-diffusion equations. Rus. Math. (Iz. VUZ).
**50**, 55–68 (2006)MathSciNetzbMATHGoogle Scholar - 7.Shishkin, G.I.: The Richardson scheme for the singularly perturbed reaction-diffusion parabolic equation in the case of a discontinuous initial condition. Comp. Math. Math. Phys.
**49**, 1348–1368 (2009)MathSciNetCrossRefzbMATHGoogle Scholar - 8.Shishkin, G.I.: Difference scheme of the solution decomposition method for a singularly perturbed parabolic reaction-diffusion equation. Rus. J. Num. Anal. Math. Mod.
**25**, 261–278 (2010)MathSciNetzbMATHGoogle Scholar - 9.Shishkin, G.I., Shishkina, L.P.: A high-order Richardson method for a quasilinear singularly perturbed elliptic reaction-diffusion equation. Diff. Eqn.
**41**, 1030–1039 (2005)MathSciNetCrossRefzbMATHGoogle Scholar - 10.Shishkin, G.I., Shishkina, L.P.: Difference Methods for Singular Perturbation Problems. Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 140. CRC, Boca Raton (2009)Google Scholar
- 11.Shishkin, G.I., Shishkina, L.P.: A higher order Richardson scheme for the singularly perturbed semilinear elliptic convection-diffusion equation. Comput. Math. Math. Phys.
**50**, 437–456 (2010)MathSciNetCrossRefzbMATHGoogle Scholar - 12.Shishkin, G.I., Shishkina, L.P.: Improved difference scheme of the solution decomposition method for a singularly perturbed reaction-diffusion equation. Trudy IMM
**16**, 255–271 (2010)zbMATHGoogle Scholar - 13.Shishkin, G.I., Shishkina, L.P.: A Richardson scheme of the decomposition method for solving singularly perturbed parabolic reaction-diffusion equation. Comput. Math. Math. Phys.
**50**, 2003–2022 (2010)MathSciNetCrossRefzbMATHGoogle Scholar - 14.Shishkin, G.I., Shishkina, L.P.: A higher order accurate solution decomposition scheme for a singularly perturbed parabolic reaction-diffusion equation. Comput. Math. Math. Phys.
**55**, 386–409 (2015)MathSciNetCrossRefzbMATHGoogle Scholar