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In this article we develop a dispersively more accurate pure advection scheme and apply it to properly simulate incompressible free surface flow equations. According to Clebsch velocity decomposition, the incompressible flow equations are decomposed into the equations accounting respectively for the flow potential, rotation, and dissipation. For the Euler equation, the midpoint implicit symplectic time integrator is applied to approximate the temporal derivative term. For the sake of reducing numerical dispersion error, the upwinding scheme is developed to minimize the difference between the numerical and exact dispersion relation equations for the time-dependent pure advection equation in wavenumber space. [ABSTRACT FROM PUBLISHER]

In this article we develop a computationally stable and dispersively accurate convective scheme for the incompressible Navier-Stokes equations predicted in non-staggered grids. To enhance the convective stability and improve the dispersion accuracy, the convective terms are approximated by conducting the dispersion analysis to minimize dispersion relation error and Fourier stability analysis. To validate the proposed third-order-accurate two-dimensional numerical scheme, we solve four problems that are all amenable to exact solutions and the lid-driven cavity problem investigated at high Reynolds numbers. Results with good rates of convergence are obtained for the scalar and Navier-Stokes problems. [ABSTRACT FROM AUTHOR]

In this paper we apply the generalized Taylor­Galerkin finite element model to simulate bore wave propagation in a domain of two dimensions. For stability and accuracy reasons, we generalize the model through the introduction of four free parameters. One set of parameters is rig- orously determined to obtain the high-order finite element solution. The other set of free parameters is determined from the underlying discrete maximum principle to obtain the monotonic solutions. The resulting two models are used in combination through the flux correct transport technique of Zalesak, thereby constructing a finite element model which has the ability to capture hydraulic dis- continuities. In addition, this paper highlights the implementation of two Krylov subspace iterative solvers, namely, the bi-conjugate gradient stabilized (Bi-CGSTAB) and the generalized minimum residual (GMRES) methods. For the sake of comparison, the multifrontal direct solver is also con- sidered. The performance characteristics of the investigated solvers are assessed using results of a standard test widely used as a benchmark in hydraulic modeling. Based on numerical results, it is shown that the present finite element method can render the technique suitable for solving shallow water equations with sharply varying solution profiles. Also, the GMRES solver is shown to have a much better convergence rate than the Bi-CGSTAB solver, thereby saving much computing time compared to the multifrontal solver. [ABSTRACT FROM AUTHOR]

Development of a stable finite element model for solving steady incompressible viscous fluid flows in three dimensions is the main theme of the present study. For stability reasons, weighting functions are designed in favor of field variables on the upstream side. For accuracy reasons, it is required that weighting functions be equipped with the streamline operator so that false diffusion errors can be largely suppressed. In the steady-state analysis of Navier­Stokes equations, we adopt the mixed formulation to preserve mass conservation on quadratic elements which accommodate the Ladyzhenskaya­Bab&ucaron;ska­Brezzi (LBB) stability condition. To resolve difficulties arising from asymmetry and indefiniteness in the resulting large-size matrix equations, we abandon the elimination-like solution solver because the storage demand exceeds the ability of our hardware to solve for three-dimensional problems. A modern iteration solver, known as the biconjugate gradiant stabilized (BICGSTAB) solution solver, is thus implemented in an element-by-element fashion to effectively alleviate the problem. For performance reasons, the finite element code developed here should be implemented in a hardware environment which is suited to the use of an iterative solver. To this end, our analysis is implemented in shared memory parallel architectures, CRAY C-90 and J-90. We benchmark the parallel computing performance through a lid-driven cavity flow problem and a problem amenable to analytic solution. [ABSTRACT FROM AUTHOR]