Energy Estimates for Discretization Errors in Water Hammer Problems

An integrated energy approach is developed for transient problems in pipelines using the fixed-grid method of characteristics. The goal is to understand how discretization errors associated with common interpolation schemes arise and how these errors can be controlled. Specifically, new analytical energy expressions demonstrate that both time-line and space-line interpolation attenuate the total energy of the system; in contrast, the wave-speed adjustment approach preserves the total energy but distorts the partitioning between kinetic and internal energy. The analytic results are confirmed through numerical studies of several series pipe systems. Both the numerical experiments and the mathematical expressions show that the discretization errors are small and can be ignored as long as the work in the system is continuous and substantial. When the work is small or negligible, however, a Courant number Cr value close to 1 is required to control the numerical dissipation.