<![CDATA[Gadunov provides insight to the general solution of Riemann problem by considering the shock as a local phenomena,and treated accordingly as local problem. Two dimensional exact analytical Riemann problems may be very difficult to solve. Another approach is to split the two dimensional wave problem into 1D Riemann problem, where the analytical solution is already known. To apply 1D Riemann problem into two dimensional wave problems, one has to discretize to the flow domain, into discrete non-overlapping elements, control volume or cells. If the elements are the control volumes, then we can employ the finite volume discretization scheme. Based on the schemes the hydrodynamic problems can be solved elegantly by transforming the domain into non-overlapping control volume elements and treated each elements as the local Riemann problem. This can be carried out by transforming the global physical coordinates into local one, foe every discrete flow elements (the control volume or cell), where the direction of flow axis is aligned with the fluxdirection. The problem is then solved as hydrodynamic fluxes, F, crossing the cells interfaces. The disadvantage of this scheme is that the control volume —representing gravitation and friction—is elimated from the equation, and therefore it seems to be not realistic. However, integration of the characteristics with the source included, it can be concluded that the source influence can be ade as small as possible by applying the proper aselection of the time step (Abbot, 1979). In this case, we are treating the problem as solving the Riemann quasi-invariants. The influence of neglecting the source term will be reflected in the amount of flux. If this influence is positive, the flux will be less than if the source term is negative, i.e the source term is sink which decreases the momentum content, otherwise it is a source which increases which increases the momentum flux. The influence must be persistence, e.g in dissipative flow regime, neglecting the source term Riemann equation will result in persistencelarger momentum flux compare to the real flow. This error may be less or greater than numerical error. (see Numerical test # 1). Therefore, when applied to the discretisized river sistem, the persistence rules will not be always reproduced in the simulation result. This alternating positive and negative effect will be reflected by plotting the numerical test value against the laboratory of field value or by complete numerical value without neglecting the gravitational and friction effects (see simulation test of Osher scheme in Kissimmee river , Florida, USA). Keywords : local problem, control volume, control volume, control volume, shock wave]]>
