Effect of Nonuniform Grids on High-Order Finite Difference Method (Advances in Applied Mathematics and Mechanics, 2018)
The finite difference (FD) method is popular in the computational fluid dynamicsand widely used in various flow simulations. Most of the FD schemes aredeveloped on the uniform Cartesian grids; however, the use of nonuniform or curvilineargrids is inevitable for adapting to the complex configurations and the coordinatetransformation is usually adopted. Therefore, the question that whether the characteristicsof the numerical schemes evaluated on the uniform grids can be preserved on thenonuniform grids arises, which is seldom discussed. Based on the one-dimensionalwave equation, this paper systematically studies the characteristics of the high-orderFD schemes on nonuniform grids, including the order of accuracy, resolution characteristicsand the numerical stability. Especially, the Fourier analysis involving themetrics is presented for the first time and the relation between the resolution of numericalschemes and the stretching ratio of grids is discussed. Analysis shows that forsmooth varying grids, these characteristics can be generally preserved after the coordinatetransformation. Numerical tests also validate our conclusions.