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When deciding whether to employ explicit or implicit approach in numerical approximation of open channel flow equations one has to bear in mind that explicit methods are generally easier to program, and require less computational effort per each time step than implicit methods. Explicit methods solve for one point at a time in certain time step - that is, one equation and one unknown. The main drawback of the explicit approach is that the unknown node must be within the zone of dependency of the nodes that are used to approximate the derivatives at the known time line – otherwise, the method will be unstable. Therefore, maximal time step is bound by Courant Friedrichs Lewy (CFL) condition. Implicit methods solve for the unknowns in a domain simultaneously, requiring solution of simultaneous algebraic equations. The advantage is that matrix of a linear system to be solved at each time level has its arguments from the new time level n+1 and thus the unknown nodes are always in the zone of influence. Generally, implicit methods also include data values from previous time level n but they come to the right hand side (RHS) of the linear system to be solved. Implicit schemes will yield convergence in a smaller number of time steps, since the time step is no longer constrained by a stability limit but on the other hand they require more computational effort consequent upon the need to solve block tridiagonal system of equations per each time step (providing finite difference approximation is employed). Use of implicit or explicit numerical method is also dependent upon the fact whether the objective of calculation is simply to reach steady state or analysis of transient solution is needed.

Claw1d 1.0 is a computer program that performs 2 tasks: numerical solution of initial – boundary value problems for one-dimensional conservation laws and solution for related optimal control problems. Three types of conservation laws are included in the present version: 1d pressurized flow equations, open channel flow equations and 1d shallow water equations. For each type of the flow model numerous boundary conditions are included. Numerical solution is based on the application of state of the art numerical methods for hyperbolic conservation law with significant source term: ENOLLF, ENO-Roe, Q-schemes, Hubbard’s, Roe’s with source term splitting. Optimal control problems that can be also solved with this program are model calibration problems, problems of finding optimal strategies for manipulations with turbines, pumps, valves, dams, weirs, etc, and optimal shape design problems. The applied optimization method is the genetic algorithm (GA). In the program many different variations of the basic GA operator are implemented. The complete program’s code is object oriented and it is written in C++. User interface is developed using the Microsoft Developer Studio Visual C++ 6.0 and it enables the user to easily build a new project, add all the components of the model, define all the necessary data, perform all computations and examine graphically and numerically all the obtained results. Graphical representations are 2d and 3d and are implemented through OpenGL. The programs is and can be applied to problems of gas-liquid mixture flows in pipelines in power plants, ships, oil industry, etc. as well as problems of flow in natural watercourses, artificial channels or propagation of flood waves.