Abstract
This article describes the development of the Hermite method of approximate particular solutions (MAPS) to solve time-dependent convection-diffusion-reaction problems. Using the Crank-Nicholson or the Adams-Moulton method, the time-dependent convection-diffusion-reaction problem is converted into time-independent convection-diffusion-reaction problems for consequent time steps. At each time step, the source term of the time-independent convection-diffusion-reaction problem is approximated by the multiquadric (MQ) particular solution of the biharmonic operator. This is inspired by the Hermite radial basis function collocation method (RBFCM) and traditional MAPS. Therefore, the resultant system matrix is symmetric. Comparisons are made for the solutions of the traditional/Hermite MAPS and RBFCM. The results demonstrate that the Hermite MAPS is the most accurate and stable one for the shape parameter. Finally, the proposed method is applied for solving a nonlinear time-dependent convection-diffusion-reaction problem.
Highlights
The convection-diffusion-reaction model is one of the most frequently used models in science and engineering [1]
The traditional/Hermite method of approximate particular solutions (MAPS) and radial basis function collocation method (RBFCM) are validated by several numerical examples
In this article, the Hermite MAPS was developed for solving time-dependent convectiondiffusion-reaction problems in two dimensions
Summary
The convection-diffusion-reaction model is one of the most frequently used models in science and engineering [1]. A meshless numerical method is proposed to solve the above-mentioned problem with improved stability and accuracy. The main focus of this study is to demonstrate the accuracy and stability improvements of the Hermite MAPS over the other three numerical methods for solving linear and nonlinear time-dependent convection-diffusion-reaction problems. The time-independent convection-diffusion-reaction problem is solved by the traditional/Hermite MAPS and RBFCM after the source term of the time-independent governing equation is computed from the results at the previous time step. We replace the Hermite RBFCM with the Hermite MAPS in the solution procedure and demonstrate that the Hermite MAPS is the most accurate and stable among these four numerical methods used for solving time-dependent convection-diffusion-reaction problems.
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