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Flood Routing by the Non-Linear Muskingum Model: Conservation of Mass and Momentum

Gąsiorowski D.

Archives of Hydro-Engineering and Environmental Mechanics

2009

Vol. 56, nr 3-4

121-137

In this paper, the conservative properties of the Muskingum equation, commonly applied to solve river flood routing, are analysed. The aim of this analysis is to explain the causes of the mass balance error, which is observed in the numerical solutions of its non-linear form. The linear Muskingum model has been considered as a semi-discrete form of the kinematic wave equation and therefore it was possible to derive its two non-linear forms. Both forms were derived directly from the kinematic wave equation. It appeared that depending on the assumed conservative form of the Muskingum equation, this model satisfies either the global mass conservation law or the global momentum conservation law. Both laws are satisfied simultaneously by the linear equation only. The mass balance error can be eliminated from the numerical solution on condition that the non-linear Muskingum equation is written in the proper conservative form.

Balance errors in numerical solutions of shallow water equations

Gąsiorowski D.

TASK Quarterly : scientific bulletin of Academic Computer Centre in Gdansk

2007

Vol. 11, No 4

329-340

An analysis of the conservative properties of shallow water equations is presented, focused on the consistency of their numerical solution with the conservation laws of mass and momentum. Two different conservative forms are considered, solved by an implicit box scheme. Theoretical analysis supported with numerical experiments is carried out for a rectangular channel and arbitrarily assumed flow conditions. The improper conservative form of the dynamic equation is shown not to guarantee a correct solution with respect to the conservation of momentum. Consequently, momentum balance errors occur in the numerical solution. These errors occur when artificial diffusion is simultaneously generated by a numerical algorithm.