Unstable Surface Modes in Finite Chain Computations: Deficiency of Reflection Coefficient Approach (Communications in Computational Physics, 2010)
In this paper, we investigate the stability for a finite harmonic lattice undera certain class of boundary conditions. A rigorous eigenvalue study clarifies that theinvalidity of Fourier modes as the basis results in the deficiency of standard reflectioncoefficient approach for stability analysis. In a certain parameter range, unstable surfacemodes exist in the form of exponential decay in space, and exponential growthin time. An approximate eigen-polynomial is proposed to ease the stability analysis.Moreover, the eigenvalues with small positive real part quantitatively explain the longtime instability in wave propagation computations. Numerical results verify the analysis.