Von Neumann stability analysis of a method of characteristics visco-elastic pipeline model

Abstract

In recent years, investigation into the transient behaviour of viscoelastic pipelines has become a focus of modelling real world systems. For a pipeline made of viscoelastic material, the circumferential stress-strain relationship of the pipe is expressed as a convolution of the pressure history with the materials creep-compliance curve in the mass continuity equation. The creep compliance curve is typically approximated by a finite number of Kelvin-Voigt models in series, each parameterised differently. This model structure lends itself to a recurrence type relationship when implemented in a method of characteristics discretised scheme. The stability of discretisation schemes is fundamental to ensure that numerical roundoff errors do not grow with time and cause large errors in the numerical results. A standard way to analyse the stability of a discretised scheme is to use the von Neumann stability analysis, which assumes a Fourier series representation of the numerical error, and studies the behaviour of the Fourier coefficients as time increases. Unstable schemes cause a growth in the coefficients, and, conversely, stable schemes result in a decay of the coefficients. This paper analyses the stability of the resulting discretised scheme when subjected to numerical errors by way of the von Neumann stability analysis. This analysis gives insight as to the relationship between the parameter ranges, of both the physical and computational systems, required for numerical stability. © BHR Group 2008.