Uniformly convergent additive schemes for 2d singularly perturbed parabolic systems of reaction-diffusion type
- 29 Downloads
In this work, we consider parabolic 2D singularly perturbed systems of reaction-diffusion type on a rectangle, in the simplest case that the diffusion parameter is the same for all equations of the system. The solution is approximated on a Shishkin mesh with two splitting or additive methods in time and standard central differences in space. It is proved that they are first-order in time and almost second-order in space uniformly convergent schemes. The additive schemes decouple the components of the vector solution at each time level of the discretization which makes the computation more efficient. Moreover, a multigrid algorithm is used to solve the resulting linear systems. Numerical results for some test problems are showed, which illustrate the theoretical results and the efficiency of the splitting and multigrid techniques.
KeywordsParabolic 2D coupled systems Reaction-diffusion Additive schemes Shishkin meshes Uniform convergence Multigrid methods
Mathematics Subject Classification (2010)65M065 65N06 65N12
Unable to display preview. Download preview PDF.
The authors thank the referees for their valuable suggestions which have helped to improve the presentation of this paper.
- 6.Farrell, P.A., Hegarty, A.F., Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: Robust Computational Techniques for Boundary Layers, Applied Mathematics, pp. 16. Chapman and Hall/CRC (2000)Google Scholar
- 7.Gaspar, F.J., Clavero, C., Lisbona, F.: Multigrid Methods and Finite Difference Schemes for 2D Singularly Perturbed Problems. Numerical Analysis and its Applications (Rousse, 2000), pp. 316–324, Lecture Notes in Comput. Sci. Springer, Berlin (1988). 2001Google Scholar
- 11.Gracia, J.L., Madden, N., Nhan, T.A.: Applying a patched mesh method to efficiently solve a singularly perturbed reaction-diffusion problem. To appear in Modeling, Simulation and Optimization of Complex Processes, Springer (proceedings of the 6th International Conference on High Performance Scientific Computing Vietnan (2015)Google Scholar