Stability Analysis of Inverse Lax-Wendroff Procedure for a High order Compact Finite Difference Schemes

This paper considers the finite difference (FD) approximations of diffusion operators and the boundary treatments for different boundary conditions. The proposed schemes have the compact form and could achieve arbitrary even order of accuracy. The main idea is to make use of the lower order compact schemes recursively, so as to obtain the high order compact schemes formally. Moreover, the schemes can be implemented efficiently by solving a series of tridiagonal systems recursively or the fast Fourier transform (FFT). With mathematical induction, the eigenvalues of the proposed differencing operators are shown to be bounded away from zero, which indicates the positive definiteness of the operators. To obtain numerical boundary conditions for the high order schemes, the simplified inverse Lax-Wendroff (SILW) procedure is adopted and the stability analysis is performed by the Godunov-Ryabenkii method and the eigenvalue spectrum visualization method. Various numerical experiments are provided to demonstrate the effectiveness and robustness of our algorithms.