1 5th International Congress of Croatian Society of Mechanics September, 21-23, 2006 Trogir/Split, Croatia IMPROVED IMPLICIT NUMERICAL SCHEME FOR ONE-DIMENSIONAL OPEN CHANNEL FLOW EQUATIONS L. Kranjčević, B. Crnković and N. Črnjarić Žic Keywords: open channel flow, implicit numerical scheme, balance laws, source term decomposi- tion, channel with irregular geometry 1. Introduction Implicit schemes are well known for the property of allowing numerical stability in the resolu- tion of partial differential equations in presence of time steps not restricted by the Courant- Friedrichs-Lewy (CFL) condition. In this work upwind implicit scheme is used for resolution of the one-dimensional open channel flow equations driven by the interest in accurate and efficient numerical modeling of nonhomogeneous hyperbolic system of conservation laws emphasizing space dependency of the flux and significant geometrical source term variation. Some research has been recently oriented to the development of implicit techniques able to capture transcritical transi- tions and discontinuities. The implicit first-order upwind scheme proposed by Yee [9] was the first non-oscillatory shock capturing implicit scheme. Some authors used implicit numerical approach in solution of open channel flow equations but not accounting for the flux space dependency [1], and in [2] even without the source splitting. Numerically balanced approximations of flux and source terms including the flux space dependency have been introduced by some authors [8], [7], [5], [4] but employing exclusively explicit numerical approach. In this paper we modified original finite volume Linearised Conservative Implicit (LCI) scheme in order to account for the spatially vari- able flux dependency, and consequently the source term was appropriately decomposed to balance the upwind flux decomposition. These numerical modifications enabled the use of implicit numeri- cal scheme in modeling of the open channel flow equations involving nonprismatic channels with rectangular cross section geometry. 2. The one-dimensional open channel flow equations The governing one-dimensional open channel flow equations [6] model the homogeneous, in- compressible, viscous water flow in rivers and channels characterized by a hydrostatic pressure dis- tribution. The one-dimensional open channel flow equations are based on conservation of mass and conservation of momentum 0 = + x Q t A ∂ ∂ ∂ ∂ (1) ( ) ( ) ( ) f b S S gA A x I g A x I g A Q x t Q − ⋅ + ⋅ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⋅ + ∂ ∂ + , , 2 1 2 ∂ ∂ (2) where t is time; x is the horizontal distance along the channel; A is the wetted cross-sectional area; Q represents discharge and g is the gravitational acceleration. The friction slope S f and the bed slope S b are defined as