This chapter will continue the subject of unsteady open channel ¢ows that was begun in Chapter 6. In Chapter 6 solutions to the St. Venant equations were obtained by making simplifying assumptions so that the C+ characteristic through the origin of the xt plane was a straight line, i.e., a uniform ¢ow existed in the channel, at time t, and the difference between the slope of the channel bottom and the energy line throughout the length of the channel, and through the time period of the solution were assumed to be equal, or Sf = So. These assumptions allowed solutions to be obtained from relatively simple algebraic equations that applied along the characteristic lines in the xt plane. These assumptions are restrictive, and rarely duplicate real world unsteady ¢ows. Available “closed-form” solutions to the St. Venant equations are very limited, and not of great practical use. Thus to obtain unsteady solutions to various ¢ow conditions that arise in open channel hydraulics we must turn to numerical methods. This chapter deals with numerical solutions of the St. Venant equations. The intent of this chapter is not to give an exhaustive treatment of the subject. The literature on this subject is vast and would ’ll volumes to describe the variety of, and large number of, techniques that have been proposed and used. Rather the approach is to provide the reader with a basic working knowledge of numerical methods as they apply to solving hyperbolic partial differential equations such as the continuity equation and equation of motion for unsteady open channel ¢ows that we call the St. Venant equations after the French Engineer De Saint-Venant who proposed them in 1871. There is probably no “best” method for obtaining numerical solution of these equations. Indeed the best method for one problem may prove de’cient for another problem. There is no substitute for a good knowledge on the part of the engineer who must obtain solutions to unsteady problems. It is hoped that readers of this text are not found in the camp of engineers who accept results from a computer just because it supplied answers in response to some input. Life is not so simple, especially when dealing with unsteady channel ¢ows.