Theory, analytical investigation, and performance of the complementary derivatives method for reducing reflection errors from nonuniform grid domains in finite difference methods

摘要

The central finite difference is used very often to approximate first-order differential equations, and it results in a second-order truncation error for a uniform grid size. Nonuniform grids are used for simulating structures with large aspect ratios or problems with large field gradients in order to improve computational efficiency. However, changing the grid size increases the truncation error at the interface between domains having different grid sizes. The error at the interface is manifested as a spurious reflection from the grid boundary, thus decreasing the simulation accuracy. The complementary derivatives method (CDM) was originally introduced as a robust discretization technique to eliminate any spurious errors arising from the changing grid sizes. In this paper, we review the theory of the CDM. We investigate the CDM analytically for the one-dimensional case and derive the fundamental modes of propagation in the numerical solution of the differential equation. Then, we calculate the reflection coefficient from the interface of two domains having different grid sizes with and without the CDM. Different representative numerical examples also demonstrate the efficiency of the CDM in reducing the reflection from the grid boundary and improving the simulation results in different applications.