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Mathematical Problems in Engineering
Volume 2017 (2017), Article ID 8379609, 13 pages
Research Article

A Self-Adaptive Numerical Method to Solve Convection-Dominated Diffusion Problems

Department of Thermal Science and Energy Engineering, University of Science and Technology of China, Hefei, Anhui 230026, China

Correspondence should be addressed to Zhi-Fan Liu

Received 5 April 2017; Accepted 11 June 2017; Published 17 July 2017

Academic Editor: Mohsen Sheikholeslami

Copyright © 2017 Zhi-Wei Cao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Convection-dominated diffusion problems usually develop multiscaled solutions and adaptive mesh is popular to approach high resolution numerical solutions. Most adaptive mesh methods involve complex adaptive operations that not only increase algorithmic complexity but also may introduce numerical dissipation. Hence, it is motivated in this paper to develop an adaptive mesh method which is free from complex adaptive operations. The method is developed based on a range-discrete mesh, which is uniformly distributed in the value domain and has a desirable property of self-adaptivity in the spatial domain. To solve the time-dependent problem, movement of mesh points is tracked according to the governing equation, while their values are fixed. Adaptivity of the mesh points is automatically achieved during the course of solving the discretized equation. Moreover, a singular point resulting from a nonlinear diffusive term can be maintained by treating it as a special boundary condition. Serval numerical tests are performed. Residual errors are found to be independent of the magnitude of diffusive term. The proposed method can serve as a fast and accuracy tool for assessment of propagation of steep fronts in various flow problems.