Construction of Numerical Stability Analysis in Systems and Error Estimate on Water Hammer Analysis in Pipeline Systems

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The aim of this study is to establish a unified method of calculating steady and unsteady flows using the method of characteristics with time-line interpolations, which is one of numerical methods based on the elastic water column model. To do this the interpolation errors are theoretically analyzed by investigating the relationships among the wave characteristics, grids, interpolations, and boundary conditions. The linear stability analysis of finite difference equations is extended so that it holds for a finite region with specified boundary conditions. Within the new stability analysis the polynomial transfer matrix is introduced, and the error characteristics can be analyzed by the complex amplify factors, which is the solution of the polynomial associated with the boundary conditions. The polynomial transfer matrix of the total system, which reflects the boundary conditions such as connections, branches, and so on, is easily derived by imposing the mass and energy continuity conditions. It makes us possible to exactly analyses the numerical errors in systems. The non-viscous analysis of the interpolation errors reveals that the errors increases monotonously as the frequency increases in a simple pipe while it does not necessarily hold for general pipe system such as series pipes, branching piping systems, and so on. With the viscous analysis without any interpolations the discretization errors have a possibility of over-damping of the lowest frequency mode adding to the highest frequency one, though the order of numerical accuracy does influence significantly. The error characteristics analyzed are examined rightly by the numerical simulations. The theoretical basis of the time marching approach for steady flows is also justified, which quickly damps out all frequency modes by maximizing the discretization errors, and a practical method of calculating the steady flows in large network pipelines is proposed.