A Petrov-Galerkin finite element model for one-dimensional fully non-linear and weakly dispersive wave propagation

A new finite element method is presented to solve one-dimensional depth-int egrated equations for fully non-linear and weakly dispersive waves. For spa tial integration. the Petrov-Galerkin weighted residual method is used. The weak forms of the governing equations are arranged in such a way that the shape functions can be piecewise linear, while the weighting functions are piecewise cubic with C-2-continuity. For the time integration an implicit p redictor-corrector iterative scheme is employed. Within the framework of li near theory, the accuracy of the scheme is discussed by considering the tru ncation error at a node. The leading truncation error is fourth-order in te rms of element size, Numerical stability of the scheme is also investigated . If the Courant number is less than 0.5, the scheme is unconditionally sta ble. By increasing the number of iterations and/or decreasing the element s ize, the stability characteristics are improved significantly. Both Dirichl et boundary condition (for incident waves) and Neumann boundary condition ( for a reflecting wall) are implemented. Several examples are presented to d emonstrate the range of applicabilities and the accuracy of the model. Copy right (C) 2001 John Wiley & Sons, Ltd.