Accurate and Efficient Numerical Solution for Trans-critical Steady Flow in a Channel with Variable Geometry SAEED-REZA SABBAGH-YAZDI * , BEHZAD SAEEDIFAR ** Civil Engineering Department, KN Toosi University of Technology, No.1346 Valiasr Street, 19697- Tehran IRAN * SYazdi@kntu.ac.ir **B_Saeedifard@computermail.net and NIKOS E. MASTORAKIS *** Military Insitutes of University Education (ASEI) Hellenic Naval Academy Terma Chatzikyriakou 18539, Piraues, GREECE *** mastor@wseas.org Abstract: - The goal of this work is to extend finite volume schemes to the hyperbolic balance laws with geometrical source term. Mixed regions of flow are considered when a super-critical to sub-critical transition takes place and hydraulic jumps occur. In this work, the shallow water equations are used to solve the open- channel flow. The equations are converted to discrete form using cell centre finite volume method for triangular unstructured meshes. For obtaining stable numerical solution, the biharmonic operator that can be computed in some certain computational stages, but its value adds to the equations in all computational steps. For preventing excessively long run times, local time stepping is used, whereby the computations on individual cells are advanced by their own maximum allowable time steps. Using cell center finite volume method for discrete formulations, proper algorithm is adopted for accurate numerical solution by considering grids, boundary conditions transferring information between nodes and centeroids of cells. The accuracy of the computed results is validated by modeling mixed sub and super critical flow in a channel with variable bottom elevation and width. Comparison of the computed water elevation with analytical solution obtained from the theoretical solution for the frictionless free surface flow shows encouraging agreements. Key-Words: - Cell Center Finite Volume Method; Shallow Water Equations; Open-Channel Flow; Unstructured Triangular Mesh 1 Introduction A number of problems can be identified with the software currently available, and as a result, research continues into developing better numerical techniques for computational hydraulics. There has been a growing trend in favor of Riemann based methods constructed within the finite volume framework. However, the computational cost of employing this algorithm can lead to excessively long run times, particularly when higher order mathematical models are used. This often is as a result of stability constraints placed upon explicit schemes, which require the smallest possible time step permitted throughout the grid, to be applied globally. The application of faster and more accurate numerical methods is considered by researchers therefore one possibility for improving this situation is to use local time stepping, whereby individual cells are advanced by their own maximum allowable time steps. To incorporate this concept into a transient model requires the development of a suitable integration strategy, to ensure that the solution remains accurate in time. Many techniques are available for numerical simulation work, such as Finite Difference methods (FDM), Finite Element methods (FEM), Spectral methods and Finite Volume methods (FVM). Within the context of open channel flows, earlier worked focused on the application of finite difference schemes and to some extent the finite element method. There are numerous finite difference schemes for spatial discretisation. They can be divided into two broad categories; central difference schemes and Proceedings of the 2nd IASME / WSEAS International Conference on Continuum Mechanics (CM'07), Portoroz, Slovenia, May 15-17, 2007 107