Assessment of the accuracy of the Newmark method in transient analysis of wave propagation problems

The Newmark average acceleration method is non-dissipative and unconditionally stable, but its accuracy in transient analysis of wave propagation problems depends not only on the spatial discretization but also on the temporal discretization. It has been found that the effects of spatial and temporal discretization when considered separately as commonly done, are far from adequate for most transient analysis. A better criterion manipulating the interdependent relationship between mesh size and time-step magnitude is imperative to achieve sufficiently accurate results of analysis. In this paper, the accuracy of the Newmark method is investigated by considering the two basic sources of errors, namely, numerical amplitude dissipation and velocity dispersion. The effects of both spatial and temporal discretizations are considered. A new technique to describe the characteristics of various frequency spectra is established. A criterion for mesh design and the selection of time-step magnitude is also proposed based on the combined effects of the amplitude dissipation and velocity dispersion. The efficiency and effectiveness of the proposed criterion are demonstrated using two one-dimensional wave propagation problems. A two-dimensional application shows that this criterion is equally applicable to multidimensional problems.