On the accuracy of the complex-step-finite-difference method (Q1636782)

To approximate first-order derivative by finite difference (FD), we usually use the following expressions \[ f'(x)=\frac{f(x+\Delta x)-f(x)}{\Delta x} + O(\Delta x), \quad f'(x)=\frac{f(x+\Delta x)-f(x-\Delta x)}{2\Delta x} + O(\Delta x^2). \tag{1} \] Another simple way is the complex-step (CS) method, which takes the imaginary part of an imaginary perturbation of the function \(f(x+\mathbf i\Delta x)\), i.e, \[ f'(x)=\frac{\mathbf{Im}(f(x+\mathbf i\Delta x))}{\Delta x} + O(\Delta x^2).\tag{2} \] The CS approximation avoids numerical cancellation errors inherent to the FD approximations, and it achieves \(O(\Delta x^2)\) with a single imaginary evaluation instead of two evaluations (the second one in (1)). The complex-step-finite-difference method (CSFDM) is a simple and efficient way to combine the CS approximations with any well-known numerical technique to solve wave propagation problems. In this paper, the authors perform extensive numerical tests comparing the CSFDM with the popular FD method (FDM) to solve the first- and second-order wave equations. Their results show that the CSFDM always yield more accurate fields than the FDM. More rigorous analysis of the SFDM such as a complete dispersion analysis is planned for future studies.