Uniform exponential stability of 1-d wave equation: The frequency domain method

Abstract

This paper is devoted to the uniform exponential stability of 1-d wave equation with boundary damping. For the continuous wave equation, it is exponentially stable. If the continuous system is discretized in spacial variable by finite difference, the discrete boundary damping is too weak to damp out the high frequency spurious modes introduced by the numerical scheme. That is to say the discrete system without any remedies does not possess the uniform exponential stability. To restore the uniform qualitative behavior, suitable vanishing viscosity term is introduced into the discrete system. However, the additional numerical viscosity terms and discrete boundary effects enhance the complexity of the discrete system and it is not easy to verify uniform exponential stability. To overcome these difficulties, the frequency domain method is employed to show uniform exponential stability.