On the Mass Conservation of the Four-Point Implicit Scheme

The four-point implicit scheme has long been applied in hydraulic modeling to solve the one-dimensional Saint Venant equations. Because of the advantages of numerical stability and efficiency, it has become the most widely adopted scheme for the simulation of unsteady open channel flows. In this paper, the scheme is re-analyzed using the viewpoint of the finite volume method. It is found that the mass conservation is defected at the inflow boundary if the parameter θ is chosen not equal to 0.5. The reason for this error is that the input flow in the model is discretized using the parameter θ and is different from the expected value. However, in practical applications θ > 0.5 is always used to maintain the unconditional numerical stability. Therefore the mass imbalance at boundaries introduces error to most cases. The error can be introduced either from the upstream boundary or from lateral inflow boundaries. The effect of this error is not serious when the modeling time step is very small or the inflow changes very slowly with time, which are generally not satisfied for actual highly unsteady flow simulations. Potential solutions for this issue are investigated and it is found that the only sound approach is to modify the scheme at the boundary.