Equivalent Differential Equations in Fixed‐Grid Characteristics Method

Abstract

Various numerical techniques, such as wave‐speed adjustment or interpolation, are generally required in order to apply the fixed‐grid method of characteristics to multipipe systems or to systems with variable wave speed. However, these techniques introduce into the solution unwanted side effects such as numerical attenuation and dispersion. The present paper develops the concept of an equivalent hyperbolic differential equation to study how discretization errors arise in pipeline applications for the most common interpolation techniques. In particular, it is shown that space‐line interpolation and the Holly‐Preissmann scheme are equivalent to a wave‐diffusion model with an adjusted wave speed, but that the latter method has additional source and sink terms. Further, time‐line interpolation is shown to be equivalent to a superposition of two waves with different wave speeds. In general, the equivalent hyperbolic differential equation concept evaluates the consistency of the numerical scheme, provides a mathematical description of the numerical dissipation and dispersion, gives an independent way of determining the Courant condition, allows the comparison of alternative approaches, finds the wave path, and explains why higher‐order methods should usually be avoided.