Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 8379609, 13 pages

https://doi.org/10.1155/2017/8379609

## 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.

#### Abstract

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.

#### 1. Introduction

It is well known that convection-diffusion equations arise in a variety of important science and engineering fields, for example, thermodynamics, fluid mechanics, and chemical reactions [1, 2]. In many applications recognized as convection-dominated problems, diffusion may be quite small in contrast to convection, so that the solution will develop steep moving fronts that are nearly shocks. Efficient and accurate assessment of propagations of moving fronts is important. However, standard Finite Difference Methods (FDM) tend to introduce spurious oscillations and stabilized methods often resort to upwind schemes, which introduce excessive numerical dissipations that will oversmooth the fronts [3, 4]. Consequently, an extremely fine mesh is needed to increase the resolution, making the convection-diffusion equation a most challenging one to be solved numerically.

There are mainly two lines of research in numerical methods for achieving higher resolution with limited computation cost. One is the development of higher-order methods trying to reduce the numerical diffusion, such as high-order FDM with flux limiters [5–10] and the method of characteristics [3, 11–14]. The other important approach is using adaptive meshes. That is, mesh points are concentrated in the region where the solution varies steeply and is less concentrated in slowly varying regions.

Different adaptive strategies, for example, Adaptive Mesh Refinement method and moving mesh methods, have shown success in solving convection-diffusion problems [15–22]. As an adaptive scheme is designed to yield highly nonuniform mesh, discretization of the governing equation on general meshes should be settled first [23]. As for how to locate mesh points adaptively, such methods often appeal to equidistributing a monitor function or solving mesh equations [19]. As a whole, adaptive methods generally consist of two parts: one is to locate mesh points adaptively and the other is to solve the governing equation on the nonuniform mesh. As a result, computation complexity is certainly increased. Moreover, a local interpolation is usually needed to project the information to new meshes. Such a remeshing operation often brings numerical diffusion. So, it is desirable to develop a numerical method that solves the problem adaptively but without complex adaptive operations.

In this paper, we propose a self-adaptive numerical method to capture the steep fronts accurately. We first consider the following convection-diffusion equation to illustrate the main ideas of our method [1, 2]:where the diffusive coefficient is positive. Instead of a traditional spatial-discretized mesh, the proposed method is based on a “range-discrete mesh” which is uniformly distributed in the value domain [24]. It is found that the range-discrete mesh has an adaptive nature because the density of mesh points is proportionate to the spatial gradient of the solution. During the computation process, positions of the mesh points are determined by the governing equation, while their values are fixed. In this way, movement of the mesh points not only solves the equation but also locates mesh points adaptively. In other words, the proposed method combines the two parts of an adaptive method, the adaptive strategy and solving the equation, in one procedure. Therefore, neither complex adaptive operations nor any interpolations are involved in the proposed method, leading to a simple and efficient method for convection-diffusion problems.

It must be emphasized that, with a linear diffusive coefficient, the solution of (1) is smooth, though there exist steep gradients, which is the most concerned situation in the literature. However, with a nonlinear diffusive coefficient, though small, spatial gradient of the solution can be discontinuous or even infinite [25–27]. This singularity phenomenon is common in many applications but often smeared out by traditional methods. By simply treating it as a boundary condition, such a singular point can be maintained in our method.

It should be noted that, for monotone solutions, the proposed method is similar to the method of Fayers and Sheldon [25]. Unfortunately, as the unknown is no longer single valued in their Lagrangian form, their method does not work for nonmonotone solutions. The proposed method overcomes this disadvantage by discretizing the governing equation over dynamic control volumes. Furthermore, how to deal with different kinds of boundary conditions with the proposed method is discussed.

This paper is organized as follows. Section 2 introduces the range-discrete strategy to obtain an adaptive mesh. Next, the numerical scheme based on the range-discrete mesh, including boundary conditions, will be illustrated in detail in Sections 3, 4, and 5. Serval numerical tests are conducted to validate the numerical method in Section 6 and conclusions are drawn in Section 7.

#### 2. Range-Discrete Mesh

Considering that the unknown in (1) has bounded value, that is, , a series of discrete values can be obtained as , where . Then, one can get a series of intersection points of the line sets and the curve , namely, as graphed in Figure 1. Then, with the location of each intersection known, approximation of is obtained. Thus, a mesh can be formed by treating each intersection as a mesh point.