Frequency domain analysis of time integration algorithms Original Research Article

In this paper, the discrete generalized Bodeʹs theorem is derived and applied to the transfer functions of time integration algorithms for semidiscrete finite element equations. An integral equation is obtained that has to be satisfied by discrete-time transfer functions of time integration algorithms. Subsequently, the frequency response of temporally discretized finite element equations and the achievable accuracy cannot be manipulated independently in different frequency ranges. However, there is a tradeoff in low and high frequency behavior of time integration algorithms. A characteristic gain that complements the role of the spectral radius of the amplification matrix is derived to quantify the asymptotic mode annihilation capability of the algorithms. As shown, the characteristic gain helps to compare different time integration algorithms. Numerical examples are presented to demonstrate the role of the proposed characteristic gain.