Balance errors generated by numerical diffusion in the solution of non-linear open channel flow equations

The paper concerns the untypical aspect of application of the dissipative numerical methods to solve nonlinear hyperbolic partial differential equations used in open channel hydraulics. It is shown that in some cases the numerical diffusion generated by the applied method of solution produces not only inaccurate solution but as well as a balance error. This error may occur even for an equation written in the conservative form not only in the solution containing a shock wave but also for the smooth solution with strong gradients. In order to explain this problem 1-D kinematic wave equation and 1-D system of shallow water equations, both solved by the finite-difference implicit box scheme, are considered. Accuracy analysis carried out using the modified equation approach revealed why the numerical solutions of these equations written in the conservative forms contain the balance errors. It was shown that the balance error is directly connected with the additional source term generated by the numerical diffusion during the solution of equation written in an inadequate conservative form. Consequently the balance error depends on the values of time step, spatial interval and the weighting parameters involved in the applied method of solution. (C) 2012 Elsevier B.V. All rights reserved.